Should I refrigerate hot sauce?

I mostly agree with GdD's answer, but I'll add a couple more comments.

1. This may be an obvious answer, but I'd generally follow the recommendation on the specific bottle. Some sauces will clearly state "refrigerate after opening," and others won't. The Frank's FAQ linked in the question is an example of these sorts of instructions: two specific products recommend refrigeration, but in other cases for Frank's hot sauces it isn't required ("but doing so will keep the product fresher for a longer period of time"). I've also seen an occasional product that said refrigeration "recommended" or something on the package. I'd generally assume that if a manufacturer bothers to say something about refrigeration, it's a good idea to do it. Sometimes it may be for food safety, but in other cases it may just be a product with ingredients that will lose flavor or degrade rapidly at room temperature.

2. Most hot sauces won't degrade noticeably at room temperature even over a period of several months, and many will be stable for a couple years. Others will change flavors slowly over time at room temperature, but not necessarily in a negative way. (Some sauces are deliberately "aged" or fermented at room temperature for periods from a week to several years, and sometimes these processes will continue to change the product even after it is bottled. Some sauces will undergo natural browning reactions at room temperature as well, which can alter flavor. I know some people who actually prefer hot sauce this way, though I'm personally skeptical.) If there's a "best before" date on your sauce and no instruction to refrigerate, you can probably assume the sauce is supposed to maintain quality at least that long without refrigeration. However, that doesn't guarantee that it will stay exactly the same over that period. As GdD said, refrigeration will almost always keep the sauce more stable.

Most hot sauces are pretty inhospitable to foodborne illnesses, and can safely stay in the cupboard rather than the refrigerator. Nevertheless, the flavors in the sauces will break down over time, storing them in the fridge as opposed to room temperature will slow deterioration and keep the sauces fresher longer. If you use your sauces quickly enough then you probably won't notice a difference, but if you like to keep them around for awhile it's worth the fridge space.

There is an argument that making hot sauce cold will dull the flavor, and this is true. However, given how small the amounts used are they will come up to temperature very quickly, a dash of cold sauce on hot food will be warm by the time you get it to your mouth, so it's not a good reason to keep hot sauces in the cupboard.

The only time I would store a hot sauce at room temperature is if it will thicken up too much to come out of the bottle easily (presuming it does not say it requires refrigeration of course). Then I will keep it in the cupboard (not exposed to sunlight).

Your typical hot sauce is vinegar based, and loaded with sugar and salt. These are all agents that inhibit foodborne pathogens, so there should be no issue with keeping them at room temperature. Perhaps if it were nothing more than pureed hot peppers, you would need to refrigerate, but not a hot sauce that is essentially pickled.

Personally, I do refrigerate my hot sauces, but that is only because I live in a very humid climate, and I have had mold start growing in my hot sauce bottles fairly rapidly.

I have a couple jars of pepper jelly (cranberry-pepper, pineapple-pepper, etc.) where the jars specifically say not to refrigerate after opening. The vinegar in the jelly will make the sugar crystalize when refrigerated. While it might not be harmful, it's unpleasant to look at and probably unpleasant to eat crystalized jelly. Based on this, I would say that it depends on the vinegar to sugar ratio in the hot sauces. The ones that say you can refrigerate after opening probably have a lower sugar content, whereas the ones that say don't refrigerate most likely have a higher sugar content.

Unless it is recommended, I don't refrigerate hot sauces. As Gary said, it dulls the flavor. If someone in my house puts the hot sauce in the fridge, I throw it away.

Putting even fermented sauce in the fridge will NOT kill the "culture". It will continue to ferment, just at a slower rate.