I am also somewhat claustrophobic.
I have not been in them, but would have problems with Mission Space and Rock N Roller Coaster.
The first few times I rode, I had problems with Haunted Mansion and Spaceship Earth. The reason was that the car sort of encloses you on 3 sides, as bjakmom noted. The cars are large enough for several people, but they have solid sides and a "roof". One of the things I had a problem with was the speakers are right behind your head. I don't know why, but it makes it worse. There is one point in the ride where your car turns around and you go down an incline backwards. That was a problem point for me. SE is never completely dark.
At HM, the first room you are in is a fairly large room (probably 20 feet square or even bigger), but at one point, the narrator points out that there are no windows and no doors. All the lights go out for a few seconds and then a door opens to a dim area where you will walk to board the ride. That area starts out large, but as you walk along, it gradually narrows to a hallway just large enough for people to walk single file ( the purpose for it is to get people into a single file line for boarding). That kind of threw me the first time because I didn't know it was going to happen. Once you get to the boarding area, you are in a large, dim room.
I didn't personally have a problem with claustophobia on any other rides.