No, any legitimate charge that you place on a customer's credit card will be honored by the issuing bank of that card regardless of whether the customer has paid their credit card bill.

Each time a merchant charges a customer's credit card an authorization process begins to determine, among other things, whether or not the customer's account is in good standing. During the seconds that it takes to authorize a credit card transaction the issuing bank of the credit card is contacted to determine if the customer's account is in good standing and whether or not they have a sufficient available balance of their account to pay for the transaction.

Theoretically, it may be possible that a customer's account falls into delinquency a split second after an authorization is issued. In a case like this the issuing bank would still honor the transaction, after all, the transaction was approved and therefore will be covered.

To take this topic one step further it's worth mentioning the dangers of collecting credit card slips and then actually charging the card at a later date or time. This practice is more common among businesses that travel and do not have the means to obtain a real-time authorization on site. In a situation like this the merchant will use a manual credit card imprinter to make a carbon-copy of the card number with the customer actual signature just underneath. The danger of this practice is that there is no way of knowing whether or not a customer's credit card is good or bad until the merchant can actual authorize the card. In situations like this, the merchant would not be paid if the customer's account was in poor standing because the issuing bank would decline the transaction once the merchant finally tried to charge the card.

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Summary
Each credit card transaction is subject to an authorization and approval process to determine, among other things, if a customer's account is in good standing. If a customer's has not paid there credit card bill and has not kept their account in good standing there card will be denied at the point of sale when it's run by a merchant.

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If after reading this article you still have questions about the credit card authorization process or a merchant's liability therein please submit your questions using the link above.