Please add any that I missed, but try and keep them to this season's arc. If you can actually answer any of these, have at it, but I bet they all require pure speculation and imagination on the part of the viewer, and that within the show none of this has really been answered.
The (many) unanswered questions of season 6 arc (SPOILERS):
1. Who took control of the TARDIS, and WHY (and HOW)? Easiest answer is "The Silence" or Madam Eyepatch, but neither of those were actually stated on screen. Also, if The Silence and Madam Eyepatch are trying to kill the Doctor to save the universe, then why destroy the universe by blowing up the TARDIS? Also, HOW HOW HOW could any of those guys actually gain such control of the TARDIS?
2. Why was Canton invited to witness the Doctor die? His presence in Utah in 2011 appeared to serve no purpose.
3. Why were the Silence on Earth in the first place? Was their massive occupation of an entire planet all just to get at the Doctor?
4. Why did the Silence need humanity to go to the moon?
5. What were the Silence doing on Earth that was so evil and horrible that it warranted a subliminal command to all humans of "You should kill us on sight", causing war/genocide between humanity and the Silence?
6. How did the Doctor, Amy, Rory, and River learn who and what the Silence were? At the end of "Impossible Astronaut" no one has any remaining memory of the Silence or even knows their name. Amy's cell phone photo causes the same amnesia. How do the Doctor & Company get to the point of knowing so much about the Silence (or even that they exist!) while not looking at them?
7. At the end of "Impossible Astronaut", young River is in a spacesuit and screaming "Help me!" in a room alone with the Doctor, Canton, Amy, old River, and Rory. Amy shoots young River and we end the episode. Where did young River go? How did she get away from the Doctor & Co. and why did they give up the search for her?
8. Why was River needed to kill the Doctor? In "The Wedding Of River Song" River claims the suit is operating automatically and that she can't control it. Why does a human need to be in the suit, and why does that human need to be River?
9. How did the Silence and Madam Eyepatch gain the ability to travel through time?
10. If the Doctor was really a robot replica of the Doctor, why did he begin to regenerate when he was shot?
11. If River was brainwashed from an early age to kill the Doctor, how did she break this programming? In "Let's Kill Hitler" what was it that the Doctor did to un-brainwash River and cause her to fall endlessly, hopelessly in love with him?
12. If River was shot by Hitler, and then shot by a gaggle of Nazi's, why didn't her dress have bullet holes in it and blood stains? Even though regeneration saved River's life both those times, surely the fabric doesn't regenerate with her!
13. Where did that obituary photo of The Doctor come from? The photo is clearly the Doctor wearing a cowboy hat in Utah. Who took that photo?
14. Likewise, how exactly does everyone in the universe learn that the Doctor died (presumably without regenerating)? The only people to witness his death were Amy, Rory, River, and Canton. Who spread the word throughout the universe that the Doctor lives only until his 11th regeneration, then dies forever in Utah in 2011?
15. Please see both 15a and 15b and 15c below. After getting her doctorate in the 51st century, River is kidnapped and somehow taken back in time to 2011 where she is placed in a robotic astronaut suit that forces her (clearly stated as against her will) to "kill" the Doctor.
15a. How did River get back to the 51st century after killing the Doctor? This is a young River who does not have her own vortex manipulator yet. Do the Silence take her back to the 51st century. Why?
15b. How does she end up in prison for killing the Doctor? After somehow getting from 2011 back to the 51st century, River is arrested and put in prison for killing the Doctor. Who arrested her? Who was able to find out she killed The Doctor? How does this police force know that the Doctor is both real, and really dead, and that River is the one who killed him in 2011, and all of this falls under their jurisdiction? Why didn't they arrest her when she was a graduate student?
15c. Since this police force knows everything the audience knows about who the Doctor is and even who was in the spacesuit, how come they don't know River was a prisoner and forced to kill the Doctor against her will? If I am put in a car that I have no control over, and someone else controls that car by remote and runs over a person, did I just kill that person? Doubtful.
16. How was River able to change a "fixed point in time"?
17. How did the Silence and Madam Eyepatch learn about what a "fixed point in time" is, how to create one, etc. Even though a few other folks in this universe have time travel, the only true Lords of Time are the TimeLords. This is complicated stuff they are both apparently understanding and manipulating successfully.
18. In "The Wedding Of River Song", when the Doctor and Amy take off their eyepatches, how do they remember what it is they are doing and that the Silence are coming for them? The two of them seem 100% unaffected by removing their eyepatches. So what was the point of wearing them?
19. Rory claims that if he takes off his eyepatch, he'll forget something is coming for him through the doors. All memories of the Silence return to you when you see them, so wouldn't he remember what's going on the second they broke through? Amy could have instructed him that "something" is trying to get in after his eyepatch was off, and as soon as they entered he'd remember what it was.
20. Why did the Silence try and kill Madam Eyepatch too?
21. River says that she sent a distress call and millions of people throughout time and space came to help. Where were they and what were they doing to help?
22. Why did the Doctor need to "marry" River? What was the purpose of it?
23. Is the Doctor even married to River? Does Rory proclaiming River married to a giant robot shaped like the Doctor, that the Doctor happens to be piloting, mean they are officially and forever husband and wife?
24. River knew the Doctor's name in "Forest of the Dead." River says the Doctor did not tell River his name at the wedding. How did River learn the Doctor's name?
25. Why does River ask the 10th Doctor whether he's done "Crash of the Byzantium" with her yet? Even if River thinks she has had adventures with the 10th Doctor (and she may have in-between episodes), surely she would remember that "Crash of the Byzantium" was with a different regeneration.
25a. River has pictures of all the Doctors faces. Are they out of order?
26. If the Silence have occupied Earth for centuries, and on all continents (as stated in "Day of the Moon") then why hasn't the Doctor seen them before? Surely in all his own travels he'd have spotted one ONCE!
27. The Doctor made it so that every time anyone watches the Moon Landing, they will get a powerful subliminal message to kill every Silence instantly and "on sight". Why didn't Amy, Rory, River & The Doctor instantly want to kill every Silence in "The Impossible Astronaut"? Surely they have seen the Moon Landing before!
28. When the Doctor learns that he needs to talk with Dorium, he says "Dorium is dead." So...why not get in your time machine and go back to BEFORE Dorium is dead and talk to him then? You know, like that you would do if you needed to talk to Vincent Van Gough.
29. When the Doctor tries to call the Brig and learns he's phoned AFTER the Brig has passed away, why not travel back in time to BEFORE the Brig has died? Learning he dies does not make the Doctor part of events. He knows that Vincent Van Gough has died, but he can go back and see him. Also, just because a nurse tells you the Brig didn't see the Doctor again doesn't mean she's right. Maybe she doesn't know everything! Maybe the Brig kept some things a secret.
30. If Dorium's head is alive in a crypt as a prisoner of the Headless Monks, why does the Doctor return Dorium to his underground prison? Dorium was beheaded while helping the Doctor. You'd think The Doctor would at least try and save the talking, thinking head.
31. How was Dorium's head able to talk without a voice box? Even if space magic kept his head somehow alive and looking healthy, where did the sounds that formed words come from?
32. How does Dorium know that the question is, Doctor who?
33. Why would "Doctor Who" be the oldest question in the universe? Wouldn't the oldest question be more like "Why do we have to die?" or "Is their a God and what is His address?" etc?
34. In "Closing Time" where do those photos of the children who saw the Doctor come from? Those photos are taken on the exact day the Doctor is spotted by the children (as evidenced by their clothing). Who took those pictures? How did they know the children saw the Doctor and why do they care? Lots of people have seen the Doctor, why take pictures of those kids?
35. Hasn't the Doctor himself changed a "fixed point in time" by going in the robot? How is his escaping death by means of a robot stand-in not changing a fixed point in time?
36. In "The Waters Of Mars" the Doctor boasts about how he has just changed a fixed point in time. Why wasn't Churchhill suddenly Emperor of Rome or whatever as a result?
37. When River sees the Doctor die in Utah in 2011 does she remember that it is her
38. How can River give up her regenerations at will and transfer this magic energy into the Doctor? Even if such a procedure is possible, you probably need to know how to do it and maybe even need some personal training and equipment. I believe the Master tried to steal the Doctor's regenerations in the Movie, and he needed quite the setup inside the TARDIS to do so.
39. Why does River give all her regenerations up to save the Doctor whom she has just killed? What has happened to cause her to make this decision?
40. Why does the Doctor love River? In this entire arc, I haven't seen any evidence to suggest the Doctor really loves River (a few lines of sexual innuendo don't count). If anything, he seems to love Amy, Donna, and Rose more (all three of whom have done things that bowl him over, leave him attracted to them, etc).
Did I miss anything? Feel free to add to the list, comment, try and answer the questions, etc!
Oh! Despite all this, they did finally answer one of the most important questions of the 11th Doctor's run: Is the Doctor's bow tie a clip-on or a real bow tie? Answer: It's real! Hooray!
The (many) unanswered questions of season 6 arc (SPOILERS):
1. Who took control of the TARDIS, and WHY (and HOW)? Easiest answer is "The Silence" or Madam Eyepatch, but neither of those were actually stated on screen. Also, if The Silence and Madam Eyepatch are trying to kill the Doctor to save the universe, then why destroy the universe by blowing up the TARDIS? Also, HOW HOW HOW could any of those guys actually gain such control of the TARDIS?
2. Why was Canton invited to witness the Doctor die? His presence in Utah in 2011 appeared to serve no purpose.
3. Why were the Silence on Earth in the first place? Was their massive occupation of an entire planet all just to get at the Doctor?
4. Why did the Silence need humanity to go to the moon?
5. What were the Silence doing on Earth that was so evil and horrible that it warranted a subliminal command to all humans of "You should kill us on sight", causing war/genocide between humanity and the Silence?
6. How did the Doctor, Amy, Rory, and River learn who and what the Silence were? At the end of "Impossible Astronaut" no one has any remaining memory of the Silence or even knows their name. Amy's cell phone photo causes the same amnesia. How do the Doctor & Company get to the point of knowing so much about the Silence (or even that they exist!) while not looking at them?
7. At the end of "Impossible Astronaut", young River is in a spacesuit and screaming "Help me!" in a room alone with the Doctor, Canton, Amy, old River, and Rory. Amy shoots young River and we end the episode. Where did young River go? How did she get away from the Doctor & Co. and why did they give up the search for her?
8. Why was River needed to kill the Doctor? In "The Wedding Of River Song" River claims the suit is operating automatically and that she can't control it. Why does a human need to be in the suit, and why does that human need to be River?
9. How did the Silence and Madam Eyepatch gain the ability to travel through time?
10. If the Doctor was really a robot replica of the Doctor, why did he begin to regenerate when he was shot?
11. If River was brainwashed from an early age to kill the Doctor, how did she break this programming? In "Let's Kill Hitler" what was it that the Doctor did to un-brainwash River and cause her to fall endlessly, hopelessly in love with him?
12. If River was shot by Hitler, and then shot by a gaggle of Nazi's, why didn't her dress have bullet holes in it and blood stains? Even though regeneration saved River's life both those times, surely the fabric doesn't regenerate with her!
13. Where did that obituary photo of The Doctor come from? The photo is clearly the Doctor wearing a cowboy hat in Utah. Who took that photo?
14. Likewise, how exactly does everyone in the universe learn that the Doctor died (presumably without regenerating)? The only people to witness his death were Amy, Rory, River, and Canton. Who spread the word throughout the universe that the Doctor lives only until his 11th regeneration, then dies forever in Utah in 2011?
15. Please see both 15a and 15b and 15c below. After getting her doctorate in the 51st century, River is kidnapped and somehow taken back in time to 2011 where she is placed in a robotic astronaut suit that forces her (clearly stated as against her will) to "kill" the Doctor.
15a. How did River get back to the 51st century after killing the Doctor? This is a young River who does not have her own vortex manipulator yet. Do the Silence take her back to the 51st century. Why?
15b. How does she end up in prison for killing the Doctor? After somehow getting from 2011 back to the 51st century, River is arrested and put in prison for killing the Doctor. Who arrested her? Who was able to find out she killed The Doctor? How does this police force know that the Doctor is both real, and really dead, and that River is the one who killed him in 2011, and all of this falls under their jurisdiction? Why didn't they arrest her when she was a graduate student?
15c. Since this police force knows everything the audience knows about who the Doctor is and even who was in the spacesuit, how come they don't know River was a prisoner and forced to kill the Doctor against her will? If I am put in a car that I have no control over, and someone else controls that car by remote and runs over a person, did I just kill that person? Doubtful.
16. How was River able to change a "fixed point in time"?
17. How did the Silence and Madam Eyepatch learn about what a "fixed point in time" is, how to create one, etc. Even though a few other folks in this universe have time travel, the only true Lords of Time are the TimeLords. This is complicated stuff they are both apparently understanding and manipulating successfully.
18. In "The Wedding Of River Song", when the Doctor and Amy take off their eyepatches, how do they remember what it is they are doing and that the Silence are coming for them? The two of them seem 100% unaffected by removing their eyepatches. So what was the point of wearing them?
19. Rory claims that if he takes off his eyepatch, he'll forget something is coming for him through the doors. All memories of the Silence return to you when you see them, so wouldn't he remember what's going on the second they broke through? Amy could have instructed him that "something" is trying to get in after his eyepatch was off, and as soon as they entered he'd remember what it was.
20. Why did the Silence try and kill Madam Eyepatch too?
21. River says that she sent a distress call and millions of people throughout time and space came to help. Where were they and what were they doing to help?
22. Why did the Doctor need to "marry" River? What was the purpose of it?
23. Is the Doctor even married to River? Does Rory proclaiming River married to a giant robot shaped like the Doctor, that the Doctor happens to be piloting, mean they are officially and forever husband and wife?
24. River knew the Doctor's name in "Forest of the Dead." River says the Doctor did not tell River his name at the wedding. How did River learn the Doctor's name?
25. Why does River ask the 10th Doctor whether he's done "Crash of the Byzantium" with her yet? Even if River thinks she has had adventures with the 10th Doctor (and she may have in-between episodes), surely she would remember that "Crash of the Byzantium" was with a different regeneration.
25a. River has pictures of all the Doctors faces. Are they out of order?
26. If the Silence have occupied Earth for centuries, and on all continents (as stated in "Day of the Moon") then why hasn't the Doctor seen them before? Surely in all his own travels he'd have spotted one ONCE!
27. The Doctor made it so that every time anyone watches the Moon Landing, they will get a powerful subliminal message to kill every Silence instantly and "on sight". Why didn't Amy, Rory, River & The Doctor instantly want to kill every Silence in "The Impossible Astronaut"? Surely they have seen the Moon Landing before!
28. When the Doctor learns that he needs to talk with Dorium, he says "Dorium is dead." So...why not get in your time machine and go back to BEFORE Dorium is dead and talk to him then? You know, like that you would do if you needed to talk to Vincent Van Gough.
29. When the Doctor tries to call the Brig and learns he's phoned AFTER the Brig has passed away, why not travel back in time to BEFORE the Brig has died? Learning he dies does not make the Doctor part of events. He knows that Vincent Van Gough has died, but he can go back and see him. Also, just because a nurse tells you the Brig didn't see the Doctor again doesn't mean she's right. Maybe she doesn't know everything! Maybe the Brig kept some things a secret.
30. If Dorium's head is alive in a crypt as a prisoner of the Headless Monks, why does the Doctor return Dorium to his underground prison? Dorium was beheaded while helping the Doctor. You'd think The Doctor would at least try and save the talking, thinking head.
31. How was Dorium's head able to talk without a voice box? Even if space magic kept his head somehow alive and looking healthy, where did the sounds that formed words come from?
32. How does Dorium know that the question is, Doctor who?
33. Why would "Doctor Who" be the oldest question in the universe? Wouldn't the oldest question be more like "Why do we have to die?" or "Is their a God and what is His address?" etc?
34. In "Closing Time" where do those photos of the children who saw the Doctor come from? Those photos are taken on the exact day the Doctor is spotted by the children (as evidenced by their clothing). Who took those pictures? How did they know the children saw the Doctor and why do they care? Lots of people have seen the Doctor, why take pictures of those kids?
35. Hasn't the Doctor himself changed a "fixed point in time" by going in the robot? How is his escaping death by means of a robot stand-in not changing a fixed point in time?
36. In "The Waters Of Mars" the Doctor boasts about how he has just changed a fixed point in time. Why wasn't Churchhill suddenly Emperor of Rome or whatever as a result?
37. When River sees the Doctor die in Utah in 2011 does she remember that it is her
38. How can River give up her regenerations at will and transfer this magic energy into the Doctor? Even if such a procedure is possible, you probably need to know how to do it and maybe even need some personal training and equipment. I believe the Master tried to steal the Doctor's regenerations in the Movie, and he needed quite the setup inside the TARDIS to do so.
39. Why does River give all her regenerations up to save the Doctor whom she has just killed? What has happened to cause her to make this decision?
40. Why does the Doctor love River? In this entire arc, I haven't seen any evidence to suggest the Doctor really loves River (a few lines of sexual innuendo don't count). If anything, he seems to love Amy, Donna, and Rose more (all three of whom have done things that bowl him over, leave him attracted to them, etc).
Did I miss anything? Feel free to add to the list, comment, try and answer the questions, etc!

Oh! Despite all this, they did finally answer one of the most important questions of the 11th Doctor's run: Is the Doctor's bow tie a clip-on or a real bow tie? Answer: It's real! Hooray!