I am an avid hot sauce collector. I own about 2 dozen bottles in various states of completion, some of which are a couple years old. They are currently in my refrigerator. Is that where they should be?  Frank's FAQ says yes.  This article on Chow.com says no.  What do the hot sauce experts say? 
I mostly agree with GdD's answer, but I'll add a couple more comments.   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. 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.  