Depends. Usually no, but if I've played a game for a long time and just before bed, I might dream of it. For a lack of a better term, that I am inside the game, although that makes it much more interesting than it really is. It mostly happened with well structured games, for example, turn based strategies where you repeat certain actions (for example: "move units, build, end turn, repeat") - in my dreams I am usually caught in a similar loop - either observing the game (somehow) or just doing stuff myself. It's not exactly pleasant - it's actually frustrating sometimes bordering a nightmare: I'd usually be stuck in some unsolvable endless loop and it's very uncomfortable. A frequent element is that I'm trying to accomplish a goal but each new turn undoes my progress or I'm trying to escape but if I do something to advance, then something else sets me back.
It's not a great description, I know, but trust me - it's quite annoying at best. Imagine playing, say, Bejeweled or some other similar puzzle game and you know you just HAVE to finish - think "clear the screen", yet it's in endless mode, so you literally cannot do that ever so there is literally no finishing. However, the only thing you are allowed to do is keep playing.
The frustration is sometimes enough to wake me up. Not unlike waking up from a nightmare which is why I draw the comparison before.
