I've done Splash Mountain several times in Dec, and it's never been a problem, warm or cold. But then I've never actually gotten wet on it.
Kali River Rapids is another matter. While I've never actually been on it, there is a *very* similar ride at our local amusement. Count on getting drenched, as in four hours later you will still be able to wring water out of your underwear. You don't want to do this in cold weather. Even in warm weather, do it last, wear a poncho, or bring a change of clothes.