Fruits That Should Not Be Stored in the Refrigerator Apricots, Asian pears, avocado, bananas, guava, kiwis, mangoes, melons, nectarines, papayas, passion fruit, pawpaw, peaches, pears, persimmons, pineapples, plantain, plums, starfruit, soursop, and quince will continue to ripen if left out on the counter.