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.

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.