Foods You Shouldn't Refrigerate

Potatoes: The cold temperatures in your fridge will actually turn potatoes' starch into sugars, giving it a sweet flavor when you cook them (and not in a good way). These can be stored at room temperature.

Tomatoes: If you've tried this, you probably know why Tomatoes shouldn't be stored in the fridge. They get really mushy and gross. Leave them out.

Honey: A lot of people put honey in the fridge, but this will thicken it too much, and it doesn't require refrigeration to stay good.

Onions: Onions will last longer in the fridge, but not without cost. Like tomatoes, your onions will soften a bit if you put them in the fridge, and they won't be quite as crisp. Also, they'll make your other food smell like onions. Keep them out of the fridge at room temperature, and away from your potatoes.

There is an odorless, harmless and tasteless gas that fruits and vegetables give off after they are picked, called "Ethylene": 

All fruits and vegetables produce it, but some foods produce it in greater quantities. When ethylene-producing foods are kept in close proximity with ethylene-sensitive foods, especially in a confined space (like a bag or drawer), the gas will speed up the ripening process of the other produce. Use this to your advantage if you want to speed up the ripening process of an unripe fruit, for example, by putting an apple in a bag with an unripe avocado. But if you want your already-ripe foods to last longer, remember to keep them away from ethylene-producing foods, as designated in the chart above.

Here is a simple chart I found to help you better understand which fruits and vegetables to safely store where:

some info derived from: lifehacker.com and www.sparkpeople.com