There are two different problems with the tires of a vehicle that is standing for a long time. When the tire has contact to the ground it develops a flat spot. The tire is no longer round and will not run properly. This can be remedied by regular shuttling or even better setting the vehicle on blocks to lift the tires from the ground. This problem may be lesser one as it´s easier to detect (thumping that gets faster at higher speeds). It may still be able to destabilize the vehicle.
When the tire is not exercised some chemicals that preserve it and keep it soft evaporate from the outer layers. This makes the tire harder and more brittle. This is really dangerous as the tire looses a significant amount of grip. Most drivers fail to detect this properly (the majority of drivers isn´t even aware of it). The towed vehicle may fail to negotiate an easy turn especially on wet roads. The tire should be used regularly at speed and with some load for some time at operating temperature to prevent this.