Tires wearing out on trainers with small diameter bars where the tire contacts is a well known problem. If backing off the tension doesn't help, one solution is to use old tires that you wouldn't use out on the road. For example one with too many punctures or a slit sidewall. There are tires made especially for trainers, but I usually have enough old tires to last the winter.

Also you might want to consider an old rear wheel to use only on the trainer. That way you can swap it out quickly when the weather gets better. Heck I even had a bike with a cracked frame for trainer duty at one time.

A new trainer may help. Also consider rollers as they don't wear out tires due to large cylinders touching the wheels.