Some parks are better to visit on rainy days than others. At EPCOT, the only attraction that would close due to weather is Test Track. Most of the other rides — along with plenty of restaurants and shops — are located in Future World and are relatively close together so you wouldn't have to walk far in the rain.
Everything is more spread out, so you'd be caught longer in the rain. Based on the usual summer forecast, we'd recommend a plan that includes visiting Animal Kingdom in the morning and EPCOT in the afternoon. Magic Kingdom and Disney's Hollywood Studios each have a fair amount of indoor attractions.
A notable “exception” is Tower of Terror, which can (ironically) operate through severe weather. With no lightning, its incredibly rare for heavy rain to shut down a ride for very long, if for no other reason than exceptionally heavy rain usually just can't fall that long.
What is this? The show must go on, and even when it is rainy, Disney World is open. Now, of course if Disney does happen to close for a weather event (like a hurricane) they will issue refunds then. But, if it is just a rainy day at Disney, the parks will remain open and no rain checks or refunds will be given.
Slinky Dog Dash is in Toy Story Land at Disney's Hollywood Studios. Since it's an outdoor attraction, it shuts down in heavy rain and if there's severe weather or lightning in the area. Much of the queue is exposed to the elements, and even the portion of the queue that's covered isn't enclosed. And it can get HOT.
They remain open if there is rain without lightning. Very rarely, they will close for air temperature reasons, like if the outdoor temperature won't be getting out of the 40s during the day. And of course if there are hurricane warnings, guests will be kept away from the pools.
Inclement weather may also affect entertainment—including shows and parades—in both parks. All outdoor attractions will close when lightning is observed in the area. We regularly update our Disneyland Resort Rules to ensure you have a safe and enjoyable experience.
You'll find that most Guests will continue having fun even through the storms, and most rides and attractions will keep running as long as there is no lightning and it is safe to do so. Walt Disney World Resort does not usually refund tickets due to inclement weather.
It goes without saying that on a rainy day, Space Mountain will have large crowds because all other rollercoasters are outdoors. If you see that the rain will extend throughout the day, leave Space Mountain for the end right before the park closes – lines will be shorter then!
Buy A PonchoDisney sells high quality ponchos in every one of their shops in the parks for $9. Of course, you can buy a poncho for cheaper elsewhere if you know it is going to rain. If you would rather bring a raincoat along with you, that would also work in protecting against rain.