I don't think it would hurt the plant if you just clipped them off shorter, and didn't damage the skin of the plant. You do know, of course, that with opuntias, it's the little glochids that are the really nasty ones! Those little devils are really difficult to remove if you get stuck by them!
I once got thrown by a horse and landed right in the middle of a prickly pear cactus. I was removing ... and having others remove
... glochids from various parts of my anatomy for the better part of a year.
Maybe that's why, although I love cacti and have several, I do not own, nor do I care
to own, any from the opuntia family.