Some shows and parades may be altered or cancelled in the event of rain - I recall seeing a special rainy day parade at MK because it was light enough to send characters out in Main Street vehicles but too wet to send the floats out. Other outdoor shows may be cancelled if it's a hazard to performers, or some shows have alternate choreography with less dancing so they can still perform safely on a wet stage.
Mostly, though, things remain as operational as possible, especially if there is no lightning. Bring good rain gear and make the most of it; a lot of people will leave the park or huddle under cover when it rains, leaving lines short for those willing to brave the rain to get to them.