As long as the policy was paid up to date when the car was totaled, you do not have to make any additional premium payments. When the car is totaled, your insurance company will usually send you a notice that the claim has been settled and the policy closed out. Since the car is no longer in use, it will not require continuing insurance.
On the other hand, if you intend to purchase your vehicle back as salvage, then you will still need for it to be insured. You can avoid insuring the car during the repair process, but it might be a better idea to talk to your insurance company about getting stored vehicle insurance, which does not include accidents and other claims caused by driving, but protects the car against collisions or other damages that occur while the car is parked.
Because the policy is closed when the payout is made on it, there is no reason for you to receive any additional billing from the insurance company for that car. If you have multiple vehicles or liens insurance with the same company, those other policies are not affected, and should be paid as regularly scheduled.