Some tyres react very badly to Turbo Trainers (more precisely they conduct heat to the inner tube quickly and the inner tube reacts badly).
I've had more punctures on my Turbo Trainer than on the road this past winter. I've had a tyre "bunch-up", sort of delaminate, before now.
Some Tyres seem fine.
I wouldn't be confident using my TT tyre just before a race.
I use a heavy, worn out, rear 28mm Schwalbe Marathon on my Trainer, it's been 100% reliable. The added benefit is that it adds a bit of weight/momentum to the wheel, making it all feel a bit more realistic.
Worst case is that you spend 5 minutes swapping a tyre. Doesn't everyone have an old rear wheel hanging in their garage, gathering cobwebs? That's what those Nails and screws are for in your garage, for hanging wheels (that you will never use) on.