In the Alternate Universe created in "The Wedding of River Song," Amy asked the Doctor that if time was standing still, why was he still aging? He said that time was still passing normally for him.
So I'd say yes, he ages.
There's also the bit where the Master prematurely aged Ten. That's why he looked like a little Gollum thing.
While physical aging is possible, it doesn't seem to actually kill them. They just keep shriveling up or something.