Twitter is a brand and, as such, you should follow the usage established by the brand itself. Looking at http://blog.twitter.com/, I see they describe the messages as Tweets (singular: a Tweet), with a capital T, and the action of sending them as tweeting (verb: to tweet), no capitalization.
That rule should be followed until established usage starts to differ from what the brand proposes, either because people think it's silly (like writing Yahoo! with its exclamation mark all the time) or because they think it's ugly (I would personally not use a capital T for tweets, as it disrupts the standard rules of capitalization).