How can I update an expired card with a new expiration date?

According to this Support Article, the expiration date should be updated automatically in your Wallet app. If the bank issuer does not support this automatically, you may need to remove the card and add it again. For me, all of my cards have updated the expiration date automatically.

Change or remove the payment cards that you use with Apple Pay - Apple Support

The relevant portion is here:

With Apple Pay, your payment card has a unique Device Account Number used to process payments and an expiration date associated with that Device Account Number.

If your card expires or you otherwise get a new card:

• The card number and expiration date linked to the Device Account Number should be updated automatically. 
• If your card issuer doesn’t support these updates, you might need to remove the card and add it again.

Most card issuers simply push an update to your card; there is nothing to do.

If your issuer does not, you may need to remove the card and add the new one.