Trainers where the wheel makes contact with a drum (of any type) generate a lot of heat in the tire, especially on 60min+ sessions.
The problem is compounded if your tired have any wear on them.
I’d look at getting some trainer specific tires. I’m pretty loyal to Conti tires and I’ve never had issue with their “Hometrainer” tire (I put a few hundred hours on the trainer or rollers in the winter)… Vitoria makes a similar tire. These trainer tires are built to both withstand and dissipate heat from trainer drums.
They ar won’t meant to be ridden outside so you either need a second rear wheel or to set the bike up dedicated to a trainer specifically.
If you want flexibility to have a strong tire for the trainer and the ability to ride outside I’d consider something like a Conti Gatorskin or Hardshell (my preference for rollers). Vittoria makes simile training tires that will be stronger as wel.