While you can't get anyone to take over your car payments, you can have them take out a loan for the full purchase price of the vehicle and give you the money for the car. This allows you to pay off your lender while the buyer remains responsible for repaying their loan.
You may think you can simply accept future loan payments from the individual seeking to take over your car payments or have them make out a check on your behalf, but this is a dangerous strategy. If the individual misses a loan payment, it's your credit score that takes a hit instead of theirs. Having the person refinance the loan is a more sensible and safer option.
Cars are easier to refinance if the individual has good credit, proof of income, and a physical address. Another advantage to having good credit is the interest rates are usually lower. The individual also needs to be able to insure the vehicle according to the state's minimum requirements. The insurance company may be able to shift the policy and payments to the new owner or devise another way to transfer the policy. Ask your lender if it's okay for you to have someone take over your loan payments. You should also make sure that person's name takes the place of yours on any loan documents for the car to reduce your liability.