So, you don't have the Doctor in your game, you're doing something else in the Whoniverse. But you're thinking of a special episode (a Christmas special, a series opener) and drop hints about bringing the Doctor in for one night only.
Having the PCs work with the Doctor will of course give fannish players a bit of an egoboost, but maybe they find themselves working at cross purposes instead.
But which Doctor, and to what end? Which one can you do well, which would suit the tenor of your game, which would the players geek out over?
The First Doctor would suit mysteries and sense-of-wonder fantasies, the Second would fit romps, the Third dashing space-opera adventures or UNIT stories, the Fourth a mix of comedy and horror, the Fifth dashing adventures again, the Sixth confrontations with authority, the Seventh sinister machinations, the Eighth dashing adventures with a bigger effects budget and theoretically Time War stories, the Ninth quirky romps and gloomy war stories, the Tenth dashing adventures with angst, the Eleventh a bit more comedy and horror again.
Or of course there are other Doctors. The Richard E Grant version(s), the Unbound, DIY versions, here's eleven fan film Doctors, and check out the shot at 2.51... These give you more freedom, but lack the immediate fanjoy response of the televisual ones.
Or if you really want to mess with the players' heads, imagine their first meeting with the Doctor and it's actually the Valeyard. (How? I dunno, wibbly-wobbly...) And maybe away from confrontation with a normal Doctor he's more Doctor-ish, working for the good in his own way, but a bit sinister and Seventh-like. So they have to wonder if they can trust him, which could make for an interesting dynamic.
The main question, after which Doctor to use, is how to give him enough to do without overshadowing everyone else. Let's look at two handy examples: The Wedding Of Sarah Jane Smith and Death Of The Doctor from The Sarah Jane Adventures. In both of these, he only appears around the halfway point (at the cliffhanger of part one of two) so there's plenty of time for the regulars and other guests to interact, investigate and chat. He also spends most of his appearance time split off from some of the group, so they have to carry on without him solving problems on their own. And in both cases he has a hand in saving the day, but he's not the central hero of either.