If a driver has no insurance but drives a car covered by no fault gets into an accident will the owner of the car be able to file a claim with his insurance company?

No fault laws refer to injuries only not property damage. If you have comprehensive and collision coverage on your vehicle the damage to your car will be repaired subject to your deductible, unless the driver is excluded from your policy. For the driver's injuries if he does not own a vehicle then the personal injury protection on your policy will cover his injuries, once again subject to any exclusions.