You must purchase full coverage auto insurance when you initially finance the vehicle. If you choose to downgrade to liability insurance while you still owe money on the car, you are violating the contract with your lender. That means they're legally allowed to cancel your auto loan and take the vehicle away from you.
Lenders require clients to maintain full coverage auto insurance to protect their investment. Let's say a financed vehicle still has $6,000 left on the loan and is totaled in an accident that was that driver's fault. The driver still has to pay the balance of the loan.