Whenever you are issued with a replacement for a payment card that has been added to Apple Pay, you should check with your bank / card issuer to find out whether you need to take any action:
Some card issuers may automatically associate your new card with the existing Device Account Number (created when your original card was added to your device), replacing the old card. This leaves you free to continue using the same card account via Apple Pay without needing to do anything yourself.
Other issuers may require you to remove your old card from Apple Pay and add the replacement as a new card.
Any given issuer may also take different approaches depending upon the circumstances (for example: routine card replacement vs lost card).
Before removing an old card, first check with your card issuer to learn which process applies in that instance and whether you need to do anything. Depending on your bank / card issuer, this information may also be available via their website.