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I'd bet that full time drag racers have special high-risk insurance. It's probably expensive, but maybe not since the sport is pretty safe nowadays.
If the cars are sponsored, they probably don't have regular insurance. The cars are already purchased, and the money is considered marketing expense by the sponsors. If a car is wrecked, it was just marketing money - better luck next year....just guessing.
For full time drag race cars, there is a special insurance for them - can't remember the company that offers it though. It insures the vehicles whenever they are OFF the racetrack (i.e. theft, damage during transportation from one track to the next, ...). I don't believe any insurance company will pay for damage that occurs during a race. It's just the risk you take. Part of the expense of racing.
It's an excellent question. Generally your regular coverage, both liability and collision coverage, are terminated if you are participating in a "timed or a speed event." The track will not help as you sign a waiver before entrance. So damage to the car is your problem. More importantly, if you permantly disable through an accident a 30-year-old medical doctor in the stands with 35-40 more years of earning potential before him/her you can imagine the liability implications. There are safety standards to prevent such accidents but you never know. Professional participants can purchase track insurance. At what cost I do not know.