Generally, it is not necessary to store banana cake in the refrigerator unless specific ingredients in your recipe require refrigeration or if it has perishable fillings or frostings that need to be kept cold. Banana cakes tend to have a moist texture that can be negatively affected by refrigeration, as it can dry out the cake and potentially make it stale faster. However, if you live in a hot and humid climate where food spoils quickly, then you may choose to keep your banana cake refrigerated to extend its shelf life. Just make sure to place it in an airtight container or wrap it tightly with plastic wrap before refrigerating.