Do you have to refrigerate peppers?
Nope. You sure don't. I don't know why they refrigerate them in stores, unless it's to keep the green ones from turning red or something.
In my opinion, you're better off keeping them unrefrigerated, even if you bought them from the store. Here are some reasons:
1. Refrigerated peppers are prone to developing anthracnose lesions, while unrefrigerated peppers, in my experience, are not prone to it. Anthracnose is where the peppers get flat, round, soft portions on them, which eventually start to rot.
2. If you don't refrigerate them, odds are you'll see your peppers more often, and actually eat them more often because of this.
3. They continue to ripen faster unrefrigerated. This can improve the taste, and change the color. Many store-bought peppers are not fully ripe.
4. They look nice in a fruit bowl (ideally, with just peppers in it, since bananas or apples might affect how long they keep).
5. There's a high demand for refrigerator space, and peppers don't need it.
6. You won't have to bend down, open a drawer, and shift through a bunch of vegetables to find them.
7. Unrefrigerated peppers have a richer taste, in my opinion.
On the other hand, refrigeration can maybe extend the shelf-life a bit, sometimes. The difference isn't great enough for me to suggest it, though. You might save yourself a day.
Refrigerated peppers might be juicier, and retain thicker walls.
However, if you started refrigerating your peppers, you're probably better off if you keep refrigerating those particular peppers. Pick one method for each pepper. Don't keep changing their environment, or they might spoil faster. Refrigeration in the store doesn't count. Once is fine.