If cars can burn both wood and coal, aside from whatever else comes up, they might just be able to stay viable. Maybe. Forget dirt motorcycles, since they can barely even fit a passenger, and fuel storeage would finish them.
If this is implemented with an eye toward realism... (burn gas, or propane, and maybe some manufactured fuels) cars are practically dead. Large-scale coordination might be possible in cantr, but it certainly doesn't actually exist on the level required...and gutting the only really fast form of land travel in the game won't help with that. If it's implemented with intent to make the requirement easy to meet by letting you burn anything, cars will survive in most places (they're just too useful) but it seems like the main effect other than annoying everyone would be to make life hard for car thieves.
Proposing to balance it by making cars cheaper is decidedly silly, unless our cars are coal-burners. In that case, any noticable cut to metal costs would be an enormous boon to most car-makers even with a fuel cost, unless the burn rate is extremely high. It takes a fair bit of coal to make 1000 grams of steel, and you get to keep the other raw materials. If they burn a different fuel, you're 'balancing' a requirement for a continual supply of unobtanium with a small savings in the materials you actually can afford to spend...
And just hopelessly smashing cars is not something that we can hope to deal with. Offensive warfare is already a difficult feat, if you're attacking someone who has any sort of preparation. Without large vehicles, it becomes almost unimaginable. Vehicles serve an absolutely vital role as drag-shields and moble logistics (healing food transport), but also are needed for coordination. A fleet of one-man bikes would be nearly impossible to keep together and get to a common target. With a couple of minivans, you have a chance.