I've got a Tacx trainer and had the same issue as you, infact i basically melted the road tyre on my bike after one 40 minute session, little bit of rubber have stuck to the door of the utility room that i have set the trainer up in.
During this "ride" i felt an odd slipping motion as i accelerated and i think it was this that was doing the damage...ive tightend up the pressure of the roller on the rear wheel and no further tyre damage has occured....
As Garz said a proper training tyre is probably best but unless you have a spare bike or wheel this is quite a lot of Phaff.
Anyway i would tighten up the "thingy" at the back and see how it goes.
Only negative is the training tyres are a ball ache to get on and encourage you to give popeye a run for his money with your forearms afterwards. A tip from another CC'er was using talc and that made it easier!
I also have a Tacx Flow and have done around 1000 miles of pretty rigorous sessions on it. All with a standard Lithion 2 road tyre that had done quite a few miles on the road beforehand, no problems at all on the turbo, no signs of rubber etc.
I'm enjoying my turbo training sessions, i find the Tacx Flow a great bit of kit, married it up with HRM. It's amazing how the price varied for this trainer, i purchased mine from Germany for £210 delivered, so very happy.