A service contract is not automatically part of the car. Some contracts let a new owner take over, some require approval, and some end when the title changes hands. You find out which kind you have by reading your own copy and asking the provider in writing, not by assuming the buyer will sort it out.
Before you list the car
Pull your contract and find the transfer section. Note who must be notified, in what form, and whether the new owner has to meet any eligibility rule. Then ask the provider, in writing, what it needs from you and from the buyer. Ask whether a transfer changes the term, the mileage history, or the claim history on file.
At the sale
Hand over the signed transfer form if the contract calls for one, a copy of the contract itself, and your service records. Do not let the keys change hands until you have a plan for the paperwork. A buyer who hears that the contract transfers only with approval should know that before the money moves, not after.
Right after
Confirm with the provider that the new owner is recorded and that the company has the correct name and address on file. Ask for that confirmation in writing and keep it with the contract. If the provider refuses the transfer or asks for something the contract never mentioned, put that refusal in writing too, and ask which clause it relies on.
What to keep
Keep the transfer notice, the provider's confirmation, the signed form, and the date you sent each one. If the buyer later disputes coverage, those records are what show who was responsible for what. The transfer rules are also worth comparing across providers before you buy in the first place, since a contract you cannot pass along is a different product from one you can. This site's table of how we score vehicle service contract companies explains how we weigh terms like these, but your own contract is the only document that answers the question for your car.