The only thing that really changes when you close a credit card is your ability to make new purchases on the card. Closing the credit card doesn't stop the interest from accruing.
You're still responsible for making at least the minimum payment by the due date until the balance is paid in full.
If you want to avoid paying interest or reduce the amount of interest you're paying, consider transferring your balance to a credit card with a 0% balance transfer rate. Remember that you'll only get a temporary reduction in the interest rate. Once the introductory promotional period is over, you'll once again start receiving monthly interest charges.