Our tile in crumbling off our shower walls and the old drywall needs replaced as well. I purchased a 3 piece shower surround, I'm not sure if I need to mud and tape the new drywall or since the surround will be sealed I don't need to worry about it.

You still mud and tape the drywall even thought it won't show. Even with a surround I'd still recommend concrete board or wonderboard. The surround will be water tight but can still have unseen leaks down the road.