The question is whether or not will you be charged and interest on the balance on a credit card accounts that has been closed. The answer is yes.
You will have to pay interest on any balance on your credit card even though the credit card account has been closed. The only thing that changes when you close a credit card account is that you cannot make any more charges on the credit card. You are still liable to pay off any debt of balances that you had on the credit card prior to closing.
Having a balance on a closed credit card account also affects the credit rating. When you close a credit card account the credit limit on that card is immediately reduced to 0. This decreases your total credit limit. But since you continue to have a balance on that credit card your credit utilization remains the same. This means that your credit utilization ratio will become higher when you close a credit card accounts that has a balance on it. While closing any credit card has a tendency to increase your credit utilization ratio, closing a credit card that still has a balance on it as a worse effect on the credit utilization ratio which is bad for the credit rating.