What is the best way to store LEGO rubber tires?
I can't speak to "best". But as a point of reference, many many of my lego tires are many years old and they are usually stored on rims. The older tires were often not removable from the rims. As I rarely build cars, I take no special care of the tires, and I've found that there doesn't seem to  be any particular problems. The rubber ages over time, but this is true of all rubber in all circumstances, and the storage method doesn't seem to have affected them much. I have these large tires from an old space set from 1983 



that are in excellent condition; sadly, the rims they were mounted on 

 

were fragile and many of those are broken. But the tires are okay.