I have a plastic "retainer sleeve" that threaded into my shower faucet handle a bit off and is now stuck. Any advice on getting this piece off?
Your shower faucet handle appears to be made of metal. Heat the metal by soaking it in hot water for a while. Then try unscrewing the sleeve. The heated metal should expand making the threads looser.
Be careful to avoid buring yourself with hot metal.
I've had luck in the past using a flat head screw driver or an awl to remove stuck pieces of threaded plastic. It will damage if not ruin the plastic piece, but if you drive the point in a bit and then angle the piece, you can slowly cause it to unscrew enough to get a pair of vise grips to finish it.