The extra mass of the helmets would likely lead to neck injuries unless the head is restrained and neck braces are also worn, such as in race cars. But then peripheral vision is restricted, which would likely lead to further accidents.
Do you have any specific, citable, data to suggest that head injuries from tires bursting is a real problem? And what is the leading cause of bursting tires? If tires typically burst from lack of maintenance or improper tire pressure, do you think drivers who fail to properly care for their tires are going to take the time to wear a helmet, neck brace, etc.?
-Rod