It depends how hard it rains. The rain last week shut down Radiator Springs/Big Thunder intermittently and Story Book/Alice were closed for a bit longer. Screamin', however, never closed. If its heavy rain, I'd expect most outdoor rides to close.
While rain definitely sends some people packing, there are still fewer attractions for people to stand in line for increasing wait time. If its raining before 8am, you'll probably get shorter lines, its easier to pack it in when you're not there yet. If it rains anytime past 9 when it seems most families make it in to the park, they're probably going to stick with the rain.