Man, that tire is hitting something when you are sitting on the bike. My bet it is the first crossbar, because the rubber bits are right after it on that towel. If the rubber was primarily on the back of the trainer wheel, I would say that was binding and destroying the tire, but it doesn't look like any rubber is staying on the trainer drum wheel, and there isn't a ton of rubber out the back.

I would check the arms that hold the front crossbar and see if they are warped, bent, or the screw holes are elongated enough to put your bike onto that crossbar.