As a non member of the Democratic party, my understanding is this:
Lieberman ran against the Democratic candidate in Connecticut. He said he was the "real" Democrat and called on leading party members, including Barack Obama, to support him, which they did. He was given the chairmanship of an important committee. He did nothing with it - although it was within his purview to investigate the Hurrican Katrina "response", warrantless wiretapping, etc., and although he was requested to do so, he refused, siding with the Bush administration. He voted against the Democratic caucus on key issues such as improving veteran's benefits, emergency contraception for rape victims (Liberman callously said women can just go from hospital to hospital until they find one who will provide emergency contraception, no need to require it) and other issues.
In the 2008 presidential election, he said he would endorse McCain based on longtime friendship and respect but would not campaign explicitly against the Democratic ticket, and then proceeded to do just that, saying Obama was naive, repeating the false statement that Obama voted to cut off funding for the troops, saying that he feared for the country if Obama was elected, etc. When asked if Obama was a Muslim, he did not say no, and it doesn't matter, like Colin Powell did, he said he did not "think" so. When asked if Obama was a Marxist, he said "that's a real good question". He also said he would not campaign for other Republican candidates and proceeded to do just that, on the stated grounds that a Democratic majority would be bad for the country.
Some people, not just Maddow and Oberman, feel that if left as chairman of the Homeland Security Committee, he will do to Obama what he refused to do to Bush. Every time Obama makes a controversial or disputed decision, and he will make them, Lieberman could be launching investigations, hearings, etc. against Obama. While Sen. Evin Bayh has said that if Lieberman uses his position in such a way the Democratic caucus can revoke his chairmanship, this would, politically, be tough to do as it would be construed, and depicted as, a cover-up (i.e. Lieberman investigates Obama and loses his position).
Perhaps Democrats on the board can add to this. I think this sums it up.
And it is far from "far left" who have concerns about Lieberman, not to mention the fact that Maddow and Olberman are only "far left" if you use that term for everyone who does not support George Bush from centrist Democrats to Marxists (as Talkhard used to).