What mibbit does is significantly different from what most "CRUD" type of apps do, though. I mean, you have templates/views like everyone else, controllers, but then your 'model' is an IRC server, right?
Sure, it is a special case in that there's the long lived http connections - comet if you're into buzz words.
The irc client is just a plugin to the server though. Very much a general purpose system...
As I say though I'm just not a 'framework' type of person - apart from when its my own ;)
None of them... Write your own. This is what I did for Mibbit, and it worked out very well. Of course it depends on what you're writing, and if any frameworks out there are up to the job, so maybe this advice is unhelpful for you.
You might however be surprised just how easy it is to create your own custom framework.