I thought I'd put this up here, from my own experience and others I've heard of. If you like to add anything, please do.
Visibility and sometimes movement is impared in certain costumes; think fursuits, quadsuits, winged suits etc. Be aware that the person might not see you if they're in a mask. So don't go glomping some cuddly looking Kyuubi, because he might not see you and you could hurt them. Approach from the front and make sure they see/acknowledge you.;
I've personally been glomped from behind in a costume, if wasn't nice as we were near a set of stairs and I was wearing very awkward footware. I've also had my tail tugged, which leads me to....
poking/pulling/tugging of costumes. I know a friend who had her Alien costume's tail snap off because someone tugged it. You also may have grubby fingers. Someone whose spent huge amounts of time and money on their costume doesn't want crubs or sweaty finger marks on them. If your a spotter, watch out for any little child who might try to pull at tails, cloaks etc.
I'll probably think of something else to add, and this is generally towards fursuits, masked costumes.