See this Bloomberg story for more on the matter, and on Burris. See here for Senator Reid's statement that says Burris "will not be seated by the Democratic Caucus," which I assume means the Democratic Caucus will vote not to seat him, rather than that they'll seat him in the Senate but just won't let him join the Democratic Caucus within the Senate.
If there's some evidence that Burris's appointment was indeed the result of a bribe or some illegal maneuvering, then indeed the Senate can refuse to seat him. But if there is no such evidence, then for reasons I noted earlier, I think their position is legally unsustainable, given the Supreme Court's Powell v. McCormack precedent. And if Burris is indeed clean, then I don't see much of a political upside to a Democratic refusal to seat him, either.
UPDATE: Prof. Brian Kalt (Concurring Opinions) agrees with me on the legal issue.
Some commenters also suggest that the constitutional issue may be different because Burris is being appointed, not elected (whether by the people or, as before the Seventeenth Amendment, by the state legislature). I don't see any basis in the text of the Constitution to treat lawfully appointed Senators differently from lawfully elected ones, except that the provision about "Elections [and] Returns" would presumably be read as covering an inquiry into the lawfulness of the appointment — e.g., whether it was the result of a bribe — just as it would normally cover an inquiry into the lawfulness of the election.
But if the argument is simply that Blagojevich is generally a criminal, and not that the appointment of Burris was done criminally, I don't see how that can fit within the Senate's power given -Powell v. McCormack.