Lay Off Lieberman
But much of the criticism paints his decision as wrong in itself (and so presumably it would be equally wrong for Lamont to run as an independent). There seem to be two lines of reasoning here: loyalty and democracy. The loyalty argument is easiest for me to dismiss, because I see loyalty to the party as a fairly minor virtue, if indeed it is one at all. Given that the party appears poised to reject him, I see no obligation on Lieberman's part to place the interests of the party institution above his obligations to fight for what's best for the people of Connecticut, America, and the world (though of course I think Lieberman is deeply mistaken about what's best for the people of Connecticut, America, and the world).
The democracy argument is that it's somehow undemocratic for Lieberman to continue running after losing a vote. This would be true if he were to continue to insist on being the Democratic Party's nominee. But he's running for Senator of all of Connecticut. The goal is to have the Senator with the broadest support among all the people of the state. If a candidate is solidly on one side of the political fence (as Lamont is), then the primary can serve as a useful test of popularity. But with a centrist like Lieberman, the views of the most liberal third of Connecticut's voters (the one who would participate in the primary) say little about the will of Connecticutians as a whole. It's quite plausible that 50% of the people of Connecticut want Lieberman for their senator, but that because those voters are spread out across both parties as well as the independents, he wouldn't get 50% of the votes in the Democratic primary. An independent run is the only way that a coalition like Lieberman's supporters, who don't sit neatly within the ideological ambit of either major party, would be able to make their will known.
Related is the idea that Lieberman's move somehow undermines the purpose of the primary. If you see the primaries as basically ways to reduce the number of candidates on the final ballot, this is true. Too many candidates on the ballot is confusing and -- in a system without IRV -- can lead to vote-splitting between closely allied candidates that ends up putting a person with minority support into office. But this justification works best when the primary system is open -- anyone can vote in either primary. This gives the system the flexibility to focus on the matchups that most need to be settled before the final ballot is printed. It's important to note in this regard that Lieberman vs Lamont is not a classic vote-splitting scenario, given Lieberman's centrism (as described above).
A closed primary system -- like Connecticut's -- serves a different purpose (albeit one also served by an open primary). A closed primary is an instrument of the party. It acts as a screening tool for deciding which candidate the party should throw its endorsement and resources behind. But this purpose is in no way jeopardized by Lieberman's independent run. He will make his run without the expectation of any support from the institutional apparatus of the Democratic Party. But there isn't, and shouldn't be, any rule that says only people who have a party institution behind them can run for office. In this sense, Lieberman sticking it out after he fails to get the Democratic Party's endorsement isn't much different from Lamont sticking it out after he failed to get the AFL-CIO's endorsement.