The first question here is: whether the team should have power to remove one of its team members. Usually trash talking on one of team members at the water-cooler is a combination of frustration and no power to do anything about the issue.
Anyway, considering there's some power delegated to the team, I think that's still a problem of management. It's manager who built the team in the first place. It's manager whose is paid to resolve such organizational and/or people issues. For me, what a team member should do here is to raise the manager's awareness of the problem. If it doesn't work with a direct superior one might try to go further up through the hierarchy, depending on the organization of course. Either way it should be management decision to remove someone from the team.
As the supporting behavior I'd look for actions aimed to remove a problem, not the person from the team, e.g. trying to discuss it through, find the source of the issue, try to find generally accepted way to improve things around.
By the way: there's a good Mike Cohn's post and even better discussion in comment sectionMike Cohn's post and even better discussion in comment section on removing team members.
