If you currently have a recurring payment and update your payment method, it does not automatically switch the recurring charge to the new card.
To switch out your payment method on a subscription, you'll need to cancel your old recurring payment and start a brand new subscription with your new card.
Once you've cancelled your old recurring payment, you can start a new recurring payment by going to the organization's WinRed page and giving again. If you don't know how to reach it, contact the organization or campaign you are trying to donate to.
Note: You will need to update your card if your old one expires.