Tallie prevents the deletion of any cards that have open transactions (anything on the Purchases page or on submitted or approved reports). To remove the cards, you'll need to either delete the charges or ensure any reports, with charges attached to the cards, are exported.
How to remove a credit card profile:
Step 1: Go to the Expense Reports page
Step 2: Check "Unsubmitted", "Submitted", and "Approved" to make sure all the transactions from all the cards have either been exported to the accounting system or archived (Archiving Expense Reports).
Step 3: You can delete the credit card profile now!
How to remove a single card within a feed:
You need to ensure that the charges or any reports with charges attached to the card are exported or archived.
After that, you can click on the trash icon on that card to delete it: