Some metal recycling facilities pay cash for old car batteries, but a more likely bet is an auto part shop. Many shops purchase old batteries with the idea of refurbishing them and then selling them at a profit.
If the management at the auto parts shop knows that the consumer selling a battery plans to buy a replacement there, sometimes the management offers a slightly better purchase price for the old battery to ensure both sides of the deal. In some cases, the management offers store credit in lieu of cash.
People who don't plan on selling the car battery should recycle it after they are done with it. Simply putting the battery into a trash bin creates the possibility of leaking acid into the environment. If metal recyclers in the area do not purchase batteries, the next step is either to take the battery to the recycler anyway, for proper disposal, or to contact the city or local government entity in charge of trash disposal and recycling. The representative can provide information about the proper way to dispose of a car battery. The requirements and locations often differ from one town to the next, so contacting the local government is the best way to find that facility.