Well, the first Doctor presumably aged till he was the 70(by human standards)-ish man we first saw. Tom Baker visibly ages over the course of his time, partly because his seven year run was the only one long enough for it to be really noticeable, but also because he was ill for much of his final season and that showed. And as Christopher pointed out, the seventh Doctor was noticeably older in the TV movie than he was when we last saw him before that. (And if you accept the season 6b explanation, the 2nd Doctor aged visibly after The War Games).
So yep, he ages. But incarnations don't usually last long enough for this to be noticeable, much as this complicates the already awkward issue of the Doctor's quoted ages!