I've never heard him say much on the matter. As he is a logical man, I'd say he likely does not, but he does "know" more than the average human. He once corrected Amy by claiming that some so-called "fairytale" creatures actually existed.
It depends on the Incarnation.
In "A Christmas Carol" (2010 Christmas Special), the Eleventh Doctor says this:
The Doctor: Father Christmas. Santa Claus. Or, as I’ve always known him, Jeff.
This implies that he knows him on a "personal" level, but seems more like a throw away line.
As Roger says in the comments, the Ninth Doctor claims he MIGHT be Santa Claus himself:
Rose: Look at you, beaming away like you're Father Christmas.
The Doctor: Who says I'm not? Red bicycle when you were twelve?
And finally, the Twelfth Doctor realizes he is in a dream only because Santa Claus is in the base with them. At the end, when they're flying over London, they have this exchange:
Santa Claus: You want to take the reins, Doctor?
The Doctor: You're a dream construct, currently representing either my recovering or expiring mind.
Santa Claus: Yes, but do you want a go?
The Doctor: Yeah. All right.
So we might say some incarnations of the Doctor believe in him, while others don't.
The doctor is kind of a fairytale himself. I don't think he started off believing in Santa because he was shocked when he saw Santa in the 2014 Christmas special when Santa had satsumas 😂