In The Flash, we see that Barry exists in the future- Reverse Flash comes back through time because of their conflict in the future. So clearly he exists there. But as far as I can recall they never mention that it means he still lives hundreds of years in the future.
So my question is, is this because he doesn't age, or because he's time travelled forward?
I don't think we know enough to know if Barry ages or not, but I think the premise of your question is not correct.
I know that in the source comics, Flash (the Barry Allen one) is not alive in the future that Thawne is from; he is an historical figure that Thawne studies and wants to emulate. Thawne eventually figures out (partly by accident) where and when Barry was born -- and also ends up going completely bonkers. His feud with The Flash crosses through time all over the place.
I think we are supposed to make a similar conclusion when it relates to the TV show. The reason that The Flash was in Thawne's future was because he chased Thawne there, not because he was still alive.
Note that Thawne has a newspaper article that he claims is something that happened in his timeline (the one he shows Eddie, about Iris's last name as Allen) that claims Flash disappeared during the Red Skies crisis. There's also a Flash museum, and while it's not impossible to have a museum to someone who's still alive, it would be kinda creepy.
To me, those hints are all leading to the conclusion that Barry Allen isn't alive in Thawne's time, but went there to fight him.
If you're asking if the Flash ages then the answer is yes. However, he does not age at the same rate as a human being. Due to the fact that he (and all the other speedsters with the exception of Bart Allen) is connected to the Speed Force, he actually ages slower than normal humans. The time traveling does not interfere with this. I do agree with KutuluMike when he says that The Flash just went into the future to fight Thawne and he wasn't actually alive at that time. However, with the time traveling and parallel timelines, it is a plausible theory that the Flash is always alive at some point. While he's in the future fighting Thawne, he might be stuck within the Speedforce in another, while another version of him is about to be struck by lightning (which turns out to be him from the future, in reality) which gives him is speedster powers.
Anyway, sorry for my ranting, this stuff is just exciting. To answer your question though, yes he does age but at a very slow rate.
Sources:
Comics
http://dc.wikia.com/wiki/Flash_(Barry_Allen)
Commen Sense and theoretical calculations
The "Crisis" mentioned in the newspapers in Wells' computer are only from (IIRC) about ten years in the future. Thawne traveled back in time quite often to face the Flash, in a non-linear manner, both to "the present" and Barry future. Indeed, the events we're seeing are from much later in Thawne's timeline, long after he'd become Flash's greatest enemy.
As for living in the future, in the comics, Iris West is actually FROM the future, and at the end of the original (pre-Crisis) run of the title, he moves to the future with her to retire. It's actually MUCH more complicated than that, and features one of the first long-form story arc in modern comics.
Barry traveling forward in time won’t affect his age. Traveling backward in time won’t affect his age. He is one person and traveling to a certain point in time and leaving doesn’t mean that he still exists in the time period now. He already existed in the time periods that he travels back to. Mostly as a kid. So when he travels back, he as one person travels back and does not still exist in that time. However his past self does. It’s complicated. But yes Barry does age and we are even shown this in one of the episodes. It was his birthday. And in another episode, when it was someone else’s birthday, Barry’s birthday was mentioned. However I’m not sure if he ages differently. He might, but all I know is that he most definitely does age.
