These foods should not be refrigerated because they will lose quality. Food safety will not be compromised, but the character, texture, or taste of the food will be adversely affected by chilling. In fact, in the case of some foods like tomatoes or potatoes, I think they become inedible when stored in the refrigerator.
POTATOES AND SWEET POTATOES
When potatoes are chilled, the starches in the flesh change to sugars. This will make the potatoes taste sweet when cooked.Make sure to keep potatoes and onions separate when stored.
CANNED PARMESAN CHEESE
If you buy the finely grated Parmesan cheese in the green can, take a look at the label. It says do not refrigerate - even after opening!
Honey will thicken and crystallize if stored in the refrigerator, even after opening. If this does happen, you can place the opened jar of honey in a pan of hot water and it will return to a pourable, smooth texture.
OILS EXCEPT NUT OILS
Most oils will thicken and turn cloudy if stored in the refrigerator. The exception is nut oils, which can turn rancid very quickly and should be stored chilled.
Do not refrigerate cookies if they do not contain a cream or custard filling. They will go stale much faster and lose quality quickly if stored in the refrigerator.
Read the label on the peanut butter you buy. Some fresh, organic brands do require refrigeration. But most ordinary peanut butters should be stored, tightly covered, in the pantry, even after opening.
Tomatoes do not like cold temperatures. You can store them in the refrigerator, but the cell structure will break down, resulting in an unpleasant mushy texture.
The starch in breads will change structure under refrigeration (called retrogradation), making the bread texture tougher. Breads also lose moisture faster in the dry refrigerator environment. Do not refrigerate breads until the label says to.
When chocolate is kept too cold, and is then brought to room temperature, condensation can make sugar condense on the top of the product, forming a rough surface called sugar bloom. Unless directed on the label, store chocolate, tightly wrapped, in a cool cupboard.
Do not store onions in the refrigerator. They should be stored in an open container in a cool, dry place - away from potatoes!