Actually, this interview Dick Durbin gave to the Chicago Tribune yesterday (from Turkey, no less), in which he said Burris's political future is "in question," doesn't leave much doubt: Durbin wants him gone. And considering that Durbin is the senior Senator from Illinois, not to mention the number two Democrat in the Senate (not to mention a big Obama ally), I'd say he'll probably get his way.
The question is, how? For reasons I gave yesterday, I doubt Burris is going to go on his own volition; he's going to have to be booted out. And that can happen one of two ways: either a super-majority of Senators can vote to expel him; or, more intriguingly, Illinois Governor Pat Quinn could arguably invoke the same 17th amendment powers Blago used to appoint Burris to end Burris's temporary term by calling for an immediate special election (which is what Illinois Rep Jan Schakowsky wants Quinn to do). At first glance, both of these moves would seem unlikely: the Senate's a collegial place, so it's hard to imagine two-thirds of its members voting to expel a colleague; and, while my knowledge of Con Law is a bit shaky, I have to imagine that if Quinn did try to end Burris's temporary term, Burris would almost surely file a legal challenge to stop him.
But something's got to give. Just from a point of self-interest, Illinois Democrats want to get Burris out of there as soon as possible, because the longer he hangs around, the harder it's going to be for them to keep that Senate seat in their hands. Given that, my guess is Quinn will be the one who does the dispatching.
P.S. Tom Geoghegan, the labor lawyer who's now running for Rahm Emanuel's old seat, wrote an interesting NYT op-ed about the 17th amendment last month.