You are just fine because piling the tires up high actually doesn't work in most cases. Most varieties are determinant so that farmers have a crop of same sized tubers maturing at one time and fitting inside the hilled rows that the machines make. So as long as you have around a foot of stuff over top of the seed potato your crop will be protected from greening.
If you are planting so many it would be more space efficient just to plant in the ground and either hill up soil or use straw.
I can't really imagine having 60-90 spare tires laying around.