A car? Not much. At any reasonable driving air pressure the air is doing most of the work.
A pickup/van with E or better rated tires? Eh, some but not much in proportion to the weight said tire is expected to carry, especially if the tires are anywhere near the top of their pressure range.
A skid steer or some other off highway machine with "bajillion ply" tires expressly designed to be resilient to sidewall punctures that runs low pressure because all of the suspension is in the tire? Yeah, sidewall stiffness accounts for a lot in that case.