Beyond just whether the Doctor ages, the larger question is, do Time Lords (and Ladies) in general age?
If you look at the stories set on Gallifrey, you see plenty of older Time Lords. So, unless they regenerate into old bodies, they must age.
It seems unlikely that they'd regenerate into older bodies because that defeats regeneration's purpose of extending their lives, as well as for comfort reasons alone!
Also, I seem to remember one older Time Lord talking to another older Time Lord in Deadly Assassin (I think), about a regenerating into a younger body due to aging.