Does the bank have to continue to pay interest on my CD after it matures?
Not necessarily. If you choose to roll over/renew the CD for another term, the bank may elect to pay the interest.
Let's say you haven't decided in advance. Once the CD matures, you may have a grace period, established by the bank, to decide whether to renew or withdraw the funds. The bank can continue to pay interest until you decide, but it's entirely up to the bank.
You may want to review your Deposit Account Agreement, which explains if interest is paid after maturity. You should have received this Agreement when you opened the account.