Of course not everyone knows TV only has about 30 more years to go.
Seriously it depends on some factors, but I think it will. I think we tend to overestimate how different the future will be and how much we will lose touch with the past. Literature survives for centuries, some even millennium. TV and movies are a relatively new medium, but a lot of them still remain touchstones to our culture after many decades. We are close to the 100th anniversary of such thing as Birth of a Nation and Charlie Chaplin, but those are still much viewed today. I Love Lucy is still shown everyday somewhere 50 years after its birth.
The big question now is that we are producing exponentially more new information now than just a few decades ago we will have to see how this information is stored and disseminated. Will most of it go the way of the Dodo, but the real gems will be preserved? Or will we acquire so much of it that people will become overwhelmed and toss it all in favor of new stuff every generation.
Star Trek has been influential enough on so much of the culture for so long that I think it will survive especially as more time gets on and people will want to see what the 20th century thought the 22-24th centuries would look like.