Well, most of these are from related works (books and the like), so the canoninity of them is questionable... BUT... Check the Doctor Who Wiki entry on Santa Claus, and there's a bit there...
Dodging the possibly non-canon entries.. In 'The Doctor Dances', he does hint to Rose that he may be Santa, via a bike she once got. Although this is contradicted by the later picture he later shows a child of himself, Santa and Albert Einstein at Frank Sinatra's hunting lodge from 'A Christmas Carol.'
So, the Doctor either IS him (and the photo is a trick... he IS a time traveler, after all; such tricks would be easy for him, or, optionally, it would hardly be the first time he has been seen in the company of himself in a different incarnation), or knows him. I'd lean toward Santa being some pan-dimensional being the the Doctor is friends with / aware of, myself.
As of the end of the episode 'Death in Heaven', we see what appears to be Santa entering the Tardis; that being said, everything being Timey-Wimey and the Doctor having met himself in different incarnations before, I can't see that this really preempts either of the possibilities I've mentioned. I'm still leaning toward him being a pan-dimensional being, but we'll see how this plays out. It is somewhat supported by him entering a Tardis in mid-flight; something we've seen beings like the White and Black guardians do.
Ok; having watched it, I can confirm; it changes nothing. WHY, I'll leave unspoken to avoid spoilers, but it's fairly conclusive, despite a TINY hint at the end that there could be more later.