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There are some foods that absolutely need refrigeration, and some that are better off stored outside of your fridge.
Do not refrigerate (unless cut):
Potatoes: Refrigeration makes the starches in potatoes turn to sugar – which is not a good thing, as it changes the flavor and makes the skin darken.
Onions: Separate them from the potatoes (they don’t play well together – potatoes can make onions rot) and store them where there is plenty of air circulation (no sealed bags or containers)
Garlic: Like onions, they prefer air circulation.
Tomatoes: Store them on the counter – they will turn mealy and dull in the refrigerator.
Avocado: Store on the counter until they are ripe.
Melon: Leave them on your counter to ripen and sweeten properly. The antioxidants in watermelon, cantaloupe, and honeydew prefer room temperature.
Stone fruits: Place peaches, apricots, nectarines, plums, cherries, and similar fruits on the counter, stem side down, to ripen. Once they start to get soft, move them to the refrigerator.
Papaya: These will not ripen in the refrigerator – keep them on the counter, and turn them occasionally so they ripen evenly. Once the skin turns yellow-orange and the fruit starts to soften, move them to the fridge.
Peppers: Store on the counter.
Store these fruits at room temperature until ripe: Guava, kiwi, apricots, mangoes, nectarines, passion fruit, pears, persimmons, and starfruit
Always refrigerate these foods:
Berries: All fresh berries are highly perishable and should be stored in the refrigerator in a container that allows air to circulate. Don’t wash berries until you are ready to use them – moisture will decrease their shelf life.
Grapes: Store in the refrigerator in a perforated bag, and don’t wash them until you are ready to use or eat them.
Pineapple: Despite its tough exterior, pineapples are quite delicate and bruise easily. If left on the counter more than a day or two, they will begin to ferment, and the acidity will increase. Refrigerate the whole pineapple in a perforated plastic bag until you are ready to cut it. After cutting, store in the fridge in an air-tight container.
Figs: Refrigerate in a plastic bag.
These items can go be stored either in the refrigerator or at room temperature – it is a matter of your preference: Apples, clementines, grapefruit, kumquats, lemons, limes, oranges, pomegranates, and tangerines
Sources:
Seriously, Stop Refrigerating These Foods [2]
Pineapple Selection and Storage [3]