Turns out, it's optional
When I was growing up, my parents always kept hot sauce in the refrigerator. I never once questioned their choices, even though I never liked putting cold hot sauce on steaming Chinese take-out. So imagine my shock when I recently learned that my parents were doing hot sauce storage wrong for all of these years. It turns out that you do not need to refrigerate hot sauce after opening the bottle. That's right. You can safely store hot sauce in your pantry or cabinet at room temperature for literally years. That's because hot sauce is fairly shelf-stable due to its high vinegar and relatively high salt content, both of which prevent bacteria from growing.
There are, however, some good reasons to refrigerate your hot sauce, starting with appearance. For example, some hot sauces change color over time if they're not refrigerated, like Tabasco. According to the McIlhenny Company's instructions, refrigerating your hot sauce is the best way to prevent this color change. A darkened hue doesn't mean your bottle of Tabasco has spoiled; it's actually got a shelf life of five years. But if the color of your hot sauce is important to you, pop that bottle in the fridge.
The other reason to refrigerate your hot sauce is to preserve the long-term quality. Even if it's not technically spoiled, five-year-old hot sauce that you've kept on your kitchen shelf is going to taste its age. That's why many manufacturers, including Frank's Red Hot, give "Best Enjoyed By" dates instead of expiration dates. "The date stamped on the shoulder or the bottom of the bottle is the date we recommend that you use the product by to assure the product's optimum flavor, freshness and consistent quality," they explain, adding, "The product will generally maintain good flavor quality for a few weeks after that date if refrigerated."
And some hot sauces have shorter shelf lives than others. The makers of Cholula hot sauce recommend that you use the bottle within six months of opening, while the makers of Tapatío give you about two years to use up an unrefrigerated bottle. So check the packaging of the bottle to figure out what's best for you and your hot sauce storage needs. And if you're in doubt, just start putting hot sauce on everything so you can use that bottle up as quickly as possible and don't have to worry about it spoiling. That's my hot sauce storage solution, anyway.