no and yes because he did sleep when he was 2nd doctor but this was only once but No, he himself does not sleep, as stated in one of the Tenth Doctor books. There are however, rooms for his various companions as he once had to jettison Romana's room to take off
On the other hand, the Tenth Doctor sort-of acknowledged that he has nightmares when he sleeps (in "The Next Doctor"). So this question really depends on which part of canon you work with. Or when you chose to believe the Doctor - in Torchwood S1 Jack also stated that he doesn't sleep, and a few episodes later we saw him waking from a bad dream. So maybe the Doctor just doesn't like to sleep, for fear of dreaming, and doesn't want his companions to know when he does. Or he's just exaggerating his physical superiority from "needs a nap every two weeks or so" to "never needs to sleep".