Having worn reproduction plate armor based on 15th century models a couple of times, my opinion is that (without a helmet) tumbling wouldn't be drastically affected as long as the wearer were strong enough to bear the extra weight. Anyone with better than average upper body strength should be fine. Wearing a helmet (other than maybe a "pot" helmet)severely restricts your head movement, so tumbling that might require head-tucking (forward rolls, for example) would be more awkward, but not impossible.
Well made plate armor actually provides nearly a full range a movement at all of the joints because of the placement of the leather straps that hold everything together. The leather is flexible, so the metal can move around quite a bit. If the joints don't "hit" in the right place on you, the armor is very restrictive. If they do, it's really no worse than wearing a thick winter coat, IME (in fact, a thick winter coat is probably worse).