Soles of shoes and tyres are to be treated chemically to make the surface strong to withstand the pressure/stress and force.
The material is treated to have proper and suitable friction coefficient between the ground/road and the surface.
They are treated to be hard enough to last longer, as they get worn out due to frictional heat melting them.
The tyres have to be hard to withstand pressure, but should be soft enough to be elastic enough as required. They must be resistant enough to sharp objects like sharp stones or edges of pits on roads. The tyres need to last at high temperatures also. The material of tyres should be such that at high temperatures, the physical properties do not get affected too much.