Well, ginsu, I know this is not applicable for F1, as you correctly point out, but all your problems derived from heat comes from the fact that you are "ingrained" into pneumatic rubber tires. That's why I posted the Lunar Rover picture; these tires couldn't care less about blistering or air pressure. They're made of metal and they're hollow, as I'm mooore than sure you already know. Now, I guess you can imagine a dozen ways to use the metal walls as dampeners.
Besides, you're not limited to friction: for example, out of the top of my head, what about quartz fibers? They could dissipate the energy by producing electrical currents, as the Bosch system mentioned...
