Ross Douthat argues today that Anthony Weiner shouldn't be forgiven unless he resigns his seat. Personally, I don't feel any need to either forgive him or not; I've never been a Weiner fan, and he's not my Member of the House...and I'm mostly unimpressed that Weiner has done anything significantly wrong, in any case (see Amanda Marcotte on the initial actions; on lies and cover-up, I'm not really paying close enough attention, but I'd have to see a lot there to be convinced).
At any rate, I mostly think that the people who need to forgive or not forgive Weiner are his colleagues, who may or may not feel that their inclination to trust him has been injured, and his constituents, with whom his representational relationship may or may not have been shattered.
As far as constituents are concerned, that's where I get to the Gramm example. Phil Gramm was a Democratic Member of the House from Texas. During the first years of the Reagan presidency, Speaker Tip O'Neill believed that Gramm, a member of the budget committee, had betrayed Democrats through a series of actions (Wikipedia has it wrong; it wasn't simply for how he voted on tax cuts), and tossed Gramm off that committee. In response, Gramm not only switched to the Republican Party, but resigned his seat in Congress, and then ran for and won back his House seat in the subsequent special election.
I don't recall anyone before or since who did what Gramm did, but I've always thought that it was an admirable choice on his part, and that it's a reasonable model for someone in, say, Weiner's current position. The problem is that his representational relationship with his constituents may be damaged? Then the way to repair it is to undergo another round of promises-election-explanation. Usually, that means waiting until the next election, but if there's a faster way, then I can imagine circumstances in which it makes sense for a politician to jump at the opportunity. Of course, the precise electoral rules matter, too...if all that happens is a back-room nomination followed by a special election in a lopsided partisan district, then it's not clear that anything would be accomplished. And it has to be an office from which resignation triggers a special election in the first place -- true of the House of Representatives, but not all other offices. Still, I guess I'm a bit surprised that no one has tried it since (unless I'm forgetting something?), and while it might be a fairly drastic option, I'd think that pols would want to keep it in their playbook.
Again, I'm not recommending it for Weiner right now. Just that it seems to be an underused option for politicians embroiled in scandals that don't involve a threat of indictment and for which constituents probably don't care -- but that the press won't shut up about.