UPDATE: Mediaite reports that Twitter is not blocked in the White House's New Media department, which is located in the adjacent Eisenhower Executive Office Building, rather than the White House itself. This is where updates to the @WhiteHouse account come from. Updates to @BarackObama, on the other hand, come from the Democratic National Committee.
Deputy Press Secretary Bill Burton also said that, while any restrictions are the result of record-keeping and security concerns, the Administration is looking into relaxing them.
In an interview on CSPAN Friday morning, White House Press Secretary Robert Gibbs noted that Twitter is blocked on White House computers.
"For some reason, Twitter is blocked on White House computers," he said when asked if he tweets. "I have to say, I'm on camera enough that I think people have a decent sense of what I'm doing minus Twittering."
The White House itself does have a Twitter, though there is no word on where those Tweets are being written.