Apogee Elite handpiece becoming very hot to touch when in use

As an Apogee Elite owner, and not a technician, I would say that the lenses are probably dirty, and are absorbing energy and converting it into heat.  Unexpected heat usually means laser energy being diverted away from its intended target.

All of the handpieces can be disassembled and cleaned, and if this isn't somehting you feel confident doing, there are laser mechanics who can do it for you.  

Good Luck!