MoveOn.org is probably going to actively recruit and support a leftist to challenge Lieberman for his senate seat next year. If Lieberman loses the nomination (which I assume would be next spring or early summer), it may be enough of a catalyst to persuade him to leave the 'Rats.
http://www.freerepublic.com/focus/f-news/1532828/posts
I'd bet a lot of money that Lieberman won't lose, assuming a serious candidate can be found to run against him in the first place. Moveon.org would find better use for their money running candidates against moderate Republicans like Chris Shays and Jodi Rell.