Much depends on the amount of rain, and whether it is a thunder storm. The ones that will close are with thunderstorms are:
Aladdin's Magic Carpets
Dumbo
Big Thunder Mountain Railroad
Splash Mountain
Astro Orbiter
Tom Sawyer's Island
Jungle Cruise
Cinderella's Golden Carousel
Swiss Family Robinson's Tree House
Liberty Bell Riverboat
Mad Teaparty
The Barnstormer
Main St Vehicles
Tomorrowland Indy Speedway
Epcot:
Test Track-- will close even with light to moderate rain.
Play Fountains
Friendship boats
Disney-MGM:
Indiana Jones Stunt Spectacular
Honey, I Shrunk the Kids Movie Set Adventure
Studios Backlot Tour (really has to be a BIG storm to close this one)
Beauty and the Beast, Live on Stage.
Animal Kingdom:
Camp Minnie-Mickey Character Greeting
Pocohontas and her Forest Friends
Kilomanjaro Safaris
Flights of Wonder
Kali River Rapids
The Boneyard
Tarzan Rocks (has to be big storm with winds)
Primeval Whirl
Triceratops Spin
I also believe they close both the Maharajak Jungle Trek and Pangami Forest Exploration in the case of bad storms