To remove a saved credit card from a patient's account, expand the Patient Information block > click on the Billing tab > click on the trash icon next to the card. 

More information about this method can be found here.

You can also just click the plus icon  in the upper righthand corner of the billing block (like you're adding a charge), select the Add Payment button on the right side, and then on the resulting payment window, select the radio button for Credit Card from the payment options at the bottom of the form.

Once you do this, you'll see a list of stored cards on the right with "Remove Card" options next to each card. If you don't see this option try hovering over the card number and make sure that the card doesn't belong to another patient (if a patient is sharing credit cards, you must delete it from the card-holder's profile):