While I'm an American and have never been to London, I can name a few things.
The London Eye - It's that giant ferris wheel you often see in the background in NuWho episodes that take place in modern London. It played an important role in the first NuWho episode, "Rose": it was what the Nestene Consciousness chose to use as a transmitter to activate the Autons.
The Tower of London - In NuWho, UNIT Headquarters is located beneath the Tower.
Big Ben - This clock tower is of of the most famous landmarks in London. It was destroyed by the spaceship the Slitheen sent crashing to earth in "Aliens of London".
Canary Wharf - This is where the Torchwood One tower was located, and where the big battle between the Daleks and the Cybermen took place. ("Army of Ghosts" and "Doomsday")
10 Downing Street - This is where the Prime Minister lives and works. The Slitheen took it over in "Aliens of London"/"World War Three", and it was destroyed by a missile at the end of the second episode (though it was obviously rebuilt by the time "Harold Saxon" was elected). I don't know if they allow visitors in there, but even if not you can still look at it from the outside.
And of course, after typing all of that I stumble upon a great site,
The Doctor Who Locations Guide. This site covers Doctor Who filming locations in depth, you should definitely check it out.
