I'm recently on Twitter, although I'm not sure I'm getting the most out of it. I don't live the sort of hyperactive urban young professional lifestyle that I think it's most aimed at.
On the other hand, I could see some use for it, one of which is playing a quick PBeM-style game against a friend also on Twitter. So what kind of games could work on Twitter, given its extremely low bandwidth.
Chess would probably be pretty easy. Chess notation is very efficient and there are easily portable magnetic chess sets that would be easy to use for a running game.
I think some other chess-like games might also work, although in some cases you'd probably have to leave it set up at a specific site and you might need to invent a short notation style.
Examples might include Proteus, Navia Dratp and Shogi.
If both players have the necessary books, it should be possible to play Ace of Aces or Lost Worlds, although with so many different Lost Worlds books it might be hard to find a match.
I think Diplomacy might be possible to play with a combination of Twitter and emails. It would be easy enough for a GM to take orders via Twitter and for players to negotiate via the service, but reporting adjudications would probably need an email.