The best practice is to release the tension on the tire after finishing your training session. Will it shorten your tire's lifespan? Maybe slightly, but tires are pretty resilient, which is why it may only reduce the tire's lifespan minimally.
As far as damaging the trainer itself, I would doubt that it would suffer any problems. If it did, there would be a serious issue with the trainer itself, and it would be revealed while riding it. I have never heard of this happening.
One way to lessen the irritation of loading and unloading your tire contact with the trainer is to find a method to consistently load the tire every time and commit it to memory (mine was two full turns of the tension knob after the roller contacted the tire, but every trainer is different). It can be helpful to put a mark on the tension knob to index it so you can consistently load the trainer every time. This will help with the drudgery of having to set up the trainer every ride as it makes tensioning the roller consistently much easier. And being much easier to hit a consistent pressure on the tire will hopefully lessen the irritation of doing it every time you ride.
Diving a bit deeper, it stands to reason that the excessive deformation that the roller of a wheel-on trainer imparts on the tire, when left in a static position for a long period of time, would not benefit the lifespan of the tire used on the trainer. This deflection is greater than what is seen by a tire when the bicycle it stored sitting on the ground without a rider. How much of an impact depends on many factors. These may be small, but just need to be identified:
- The quality/construction of the tire - this is a variable that just needs to be considered as a factor. Construction processes (belts, cords, etc.) vary between tires and how they react to a large, static deflection over a long time will vary because of this.
- The air pressure in the tire - if the air pressure is lower, the stresses at the static deflection will be lower as well. Think of a flat tire as an example.
- The diameter of the roller on the trainer - this is a more significant variable. A roller with a smaller diameter would need to deflect a tire more to have enough friction to not allow the tire to slip on the roller than a roller of a larger diameter. It comes down to friction, which is a function of force and surface area. The surface area, i.e., contact patch, of the smaller roller would be smaller than a larger roller. To achieve the same friction to avoid tire-slippage on the trainer, the force (tire deflection) needed on a smaller roller would need to be higher than that of a larger roller.
- The time the tire is deformed in one spot - tires are pretty resilient, but given enough time, they will succumb to the static force placed in one spot on their circumference. A case in point can be seen in cars/trucks. If a vehicle is not moved for a few years, the tires can be permanently flat-spotted. Long-term storage of cars/trucks recommend supporting the vehicle by its frame, removing the load on the tires and the suspension components as well. Fortunately, this is a very long time period and as mentioned the resiliency of tires will allow them to "recover" if the time period is not too long.
- The environmental conditions while the tire deformed in one spot - another variable to be considered. Temperature, humidity, sun exposure, and air quality can have some impact as well. A consistent middling temperature (room temperature), limited sun exposure, and air that is low in contaminants that would degrade a tire (i.e., a corrosive environment) would minimize the impact on the tire. I did not elaborate on humidity, but I would guess that nominal humidity would be ideal here as well.
It has been a few years since I have ridden a wheel-on trainer, but my practice had been to release the tension after every ride. Occasionally I would forget to do this in my post-ride routine, and if it had been a few days since I last rode, I would notice the flat-spot on my rear tire when riding again. Fortunately, the flat spot would go away after about a minute, no apparent harm done by leaving the tire compressed by the trainer roller on occasion.
