Answer: Yes, a tweet can be protected by copyright.
This is a question that’s been in the news recently, after the text of a shirt worn by Frank Ocean and sold by an online retailer was discovered to have been copied from someone’s tweet without authorization.
Another outstanding podcast from Portia Sabin’s Future of What podcast, this time with the music credits services Jaxsta and Discogs on the importance of credits. Credits are also known as the moral right of attribution, currently under review at the U.S. Copyright Office. Yet another artist rights area where the U.S. lags behind the rest of the world. (See also Robert Levine’s recent post on Billboard about Auddly, a different and also important approach to the credit issue and MTP’s previous post on how Facebook sell artist names as keywords without rights.)