I don't think that they would explode.
Car tires do have a pressure of 2-3 atmospheres in normal atmosphere. It's not a big difference to 2-3 atmos against vacuum.
A bigger problem would be the extreme heat from the sunlight.
I'm not sure, but i don't think that the tires of the spaceshuttle are stowed in the gearbox under pressure. But at least
they are hidden from the suns influence.
I would say even a coke-bottle would'd explode in vacuum as long as the plastic doesn't melt

But another question is bugging me for a long time: Could a small hole in the wall of a spaceship stuffed by a thumb if it's
smaller than 1 cm²? The pressure on 1cm² is 1kg at 1 atmosphere. So it should be possible for a short time. But if the hole
faces the sun the thumb will get a really bad sunburn
