There are very few rides the dog cannot go on. I've taken our dog on almost everything. In MK dogs cannot go on Thunder Mountain, Splash Mountain, Space Mountain, Goofy's Barnstormer, or (of all things) Peter Pan's magic flight. To the best of my knowledge, everything else is fair game. There are a few that say to review with a CM (Buzz, Stitch, maybe a few more) but I've never not gone on them.
Epcot is Test Track and Soarin, I think that's it.
AK is Everest and (I think) A Bug's Life, that's all I know of (we rarely go there though, so I'm not positive). Animals on the safari jeep must be secured secondarily to the jeep, in addition to the owner. They're leashed to the wheel chair tie-downs.
I don't know about HS because we don't ever go.
Basically anything that requires a harness, belt, bar, or something else to restrain the rider is generally off limits to the dog. Things that actually hold the rider in, not just go in front of the rider (like haunted mansion or Nemo).
If my dog drools on something, I'll wipe it up. I don't wipe down everything he touches and don't intend to do so. My kid is also highly allergic to many things in addition to IC, I clean his environment, I don't trust or assume someone did it before me.