If you store unopened jars of food in the fridge and then move it to dry storage will the food inside become spoiled?
Not if it was properly processed for room temperature storage in the first place.
i.e. there are "refrigerator pickle" and "refrigerator jam" recipes that are never intended to be stored at room temperature, and do not have appropriate recipes or processing for room temperature storage. Those would not be safe.
But something like a jar of applesauce that was processed for room temperature storage, stored cold in the fridge, then stored warm, without ever being opened, would be fine.