The economy likely will show at least some improvement, unless we do a Greece. So Obama is going to take credit for any improvement.
If the Republicans break their no-tax pledge, and the economy improves, Obama will say that his policies worked, but the Republican base will know they didn't and be furious with Republican congressmen for raising taxes.
If the Republicans hold firm and don't raise taxes, they'll be held responsible if the economy only improves a little. Obama and the media would say that we would have a full recovery if the Republicans had cooperated.
I've read, and sometimes agree with the argument, that the Republicans should give Obama every last thing he wants. But they should be very vocal and very public, emphasizing that they are doing exactly what Obama wants, and that citizens should remember that, and the Republicans will remind them of that, in a year or two. That assumes that the media will cover things fairly, though, so the idea won't work.