Do I have to accept a store credit if the store refund policy states that no cash refunds will be issued?

There is no law that requires a merchant to refund money. Refunds are subject to the established store refund policy at the time of purchase, unless the product purchased is found to be unfit for the purpose of which it was intended. A customer changing their mind after purchase, deciding they dislike a style, color, or fit, for example, is not the fault of the merchant and the merchant generally cannot be held responsible.

When stores do refund money, it is usually pursuant to a store policy that states explicitly that returns are extended, born out of the desire to create and keep good will in the community; but, this is a store policy and not a law.

In addition, whether or not the store offers only credit to purchase items within their store, or whether to offer cash back or refund to a credit card, is also up to the company decision-makers, and can be changed at any time. However, if at the time a customer buys an item, the refund policy states that returns are permitted for cash refunds within 30 days, and then the store changes their return policy a few days later to prohibit returns, the merchant must still allow the customer to return the item within 30 days of purchase. This is because a contract is technically established at the time of purchase, which in this case states that the customer can return bought items for a refund. If a merchant denies a consumer a return when there was a permitted return policy at the time of purchase, that consumer may have grounds to file a claim in small claims court. 

When making large purchases, particularly for items such as furniture, which may be difficult to judge until you have brought the items home, it is always best to check into a store's refund policy before making the purchase. Otherwise, if the item does not fit through a doorway, for example, the buyer may be stuck with an item they cannot use, or stuck with a store credit as the only option.