If it was parked on a driveway (or at least not on the public highway, then yes they will, but if the car was on the public highway, the car was effectively being kept there illegally, and the insurance will not pay out. It would be worth checking the small print of the policy, as some insurers will invalidate the insurance if the tax, or mot have expired, irrespective of where the car was being kept. The only exclusion to this is where the vehicle is being taken to the mot centre for its test. If it is involved in an accident on the way, your cover should still be valid.
Incidently, if the insurance was invalidated, the insurance company will not give any refund for that time. You will have paid for nothing.