He does and has occasionally been seen on screen to do so. However, he seems to sleep much less than humans do.
In the audio drama "Davros" (audio story), the Doctor said that his Time Lord constitution of sleep is about an hour or so a night.
It seems like some incarnations sleep more than others. The 4th Doctor famously said, "Sleep is for tortoises", and the only time he was ever seen sleeping was when drugged (e.g., in "The Brain of Morbius"). The 11th Doctor has hinted that he may not even have a bedroom, and the Night and the Doctor DVD mini-episodes revealed that while his companions are asleep, he regularly goes on solo adventures and outings with River Song. On the other hand, the 10th Doctor slept extensively post-regeneration (as the 5th did before him), and seemed to have a collection of sleepwear, and the 7th Doctor commented that he enjoyed sleep. At the end of the 50th Anniversary Special "The Day of the Doctor", the 11th Doctor remarks that he dreams "about where [he's] going." The 12th Doctor also slept post-regeneration but only after Madame Vastra knocked him out.
The novels, if you take them as canon, give us more detail--both because sleeping isn't usually interesting enough to show on TV (and it's worse in an audio play), and because there's a lot more room to fill in 60 novels than in 12 TV serials. The 7th Doctor at one point says that he hates losing consciousness because the whole point of being alseep, possibly the whole point of being alive, is the dreams. The 8th Doctor early on mentions that he doesn't seem to like sleeping as much as his previous incarnation, and his companion Sam tells people that he never sleeps. Later, when stuck on Earth with amnesia and no idea why he's different from everyone else, he sleeps for 8 hours per night for a few years until he realizes he doesn't need to. When he meets Miranda (his future adopted daughter, and probably also the biological daughter of either a future incarnation or a later descendant), they talk about only needing a couple hours of sleep per night.