Banana
Bananas are grown in hot climates and retain their nutrients better when kept at temperature. Putting them within the refrigerator also will hamper the ripening process that happens naturally. Cell damage may even occur when bananas are kept at colder temperatures.
Apples
Most kinds of fruits are better when kept out of the refrigerator. Flavor and texture are getting to be better maintained when kept at temperatures that are warmer than refrigeration. Apples will usually last 1 or 2 weeks when kept at temperature.