Similarly anything that rotates has to withstand torque and centrifugal force. Eg wheels. (See also this: https://www.youtube.com/watch?v=K_ImAm1WhcY for what it takes for going 3x faster.)
These advances result in capital investment for R&D, percolate through the supply chain, and this can lead to better techniques here and there. (But I think after a certain point the disparity between the technology - the trade offs - needed for 100 mph and 300 mph become so large, that the resulting new/better capabilities doesn't mean anything for the normal automobile ecosystem.)
Such development might help "normal" tires improve in durability.
I can imagine this is something that happens in a lot of area's. Pushing every part to the extremes might help make them more reliable under normal usage.
Ultra high speeds also require efficient aerodynamics, which in turn helps reduce drag/fuel usage.