The current GOP plan appears to be to pass a 20 month abortion ban to appease the members who are trying to use Planned Parenthood funding to shut down the government. Obama would almost surely veto a 20 month abortion ban. Then what?
Probably nothing.
Basically, the situation is that Boehner and McConnell don't want to shut down the government, and obviously neither does Obama, Pelosi, or Reid. Between them they control all the veto points and plenty of votes, so they should be able to keep the government open fairly easily.
The problem, of course, is that part of John Boehner's caucus wants to shut down the government. Theoretically they want to shut it down to get rid of Planned Parenthood, but the reality is that they mostly just want to fight to the death, and shutting down the government is part of that. There are probably a half-dozen Representatives who would be okay with shutting the government down forever, really, so it's easy for them to want to shut down the government in any individual situation.
The percentage of Tea Partiers in the GOP House is small. But there are enough of them to do three things -- require Democratic support to pass a CR, prevent Boehner's election as Speaker without Democratic support, and scare the other Republicans into being afraid to take votes that will be read as giving in to the Democrats.
Boehner doesn't need the Tea Partiers to keep the government open. The question is whether he needs them to keep his job. So what he wants to do, and what he's generally done, is give them something -- something symbolic, like a vote on an abortion ban, or something minor, like a brief shutdown to prove that it's tactically unsound -- and then go ahead and do the thing he was always going to do.
So a plan where Boehner allows a vote on a 20-week abortion ban in exchange for allowing a vote on a continuing resolution is a win for a lot of people. Everybody gets to keep the government open, and people who want to keep the government open but want to show fight get to show fight. People who really do want to shut everything down lose, but they SHOULD lose.
The question is whether this will be sufficient to pacify the large percentage of the GOP caucus that wants to keep the government open while still being able to defend themselves against attacks from the right.