In the end, the goal should never be "lets get rid of every single car", but rather "lets shift from a Car-centric design, to a active transportation & healthy living-centric design"
I think the transition away from cars will be slow enough that this wont be a problem. In the ideal situation, people will just stop replacing their broken down cars.
We already deal with it somehow. Scrap them for metal, recycle rubber and probably just throw away plastic, because plastic recycling is pretty crappy.