It depends on what you're speaking of. From a food safety perspective soy sauce, ketchups, prepared mustards and Worcestershire, along with vinegars and prepared hot sauces like Tabasco are shelf-stable. Peanut butter, unless one goes through the jar in a matter of days or unless it contains preservatives, should be fridged as oils rancidify quicker at room temp and with exposure to light.
From the same perspective: Many commercial Q sauces, hoisin and oyster sauces, ketchups, jams and preserves contain acids and high concentrations of sugars. Often, neither the acid content nor sugar concentration is high enough for true shelf stability though it depends on the product, how it was manufactured, and whether it contains chemical preservatives. Spoilage bacteria (which are different from the bacteria discussed in a food safety context) and mold spores might establish and develop but, again, this depends on product specifics. It also depend on how quickly a product is used. Ketchup, e.g., while fairly stable, is used quickly enough in diners and many restaurants that spoilage is not a concern. In fine dining establishments--and most homes, I'd say--it is not and so would be subject to spoilage.
Most important from my perspective, however, is the effect of air, room temps, and light on products that contain high concentrations of flavor volatiles in suspension, which many of these prducts contain. Some contain items (tomato and mustard, e.g.) whose flavors are better at room temp or warm. But most important to me is preservation of all of these flavor components. Thus, because I do not go through any of these products all that fast and flavor preservation is of prime importance to me, I refrigerate all of products noted in this post, save for a tiny refillable bottle of hot sauce that camps on my table. I have never bought commercial Q sauce but several people have sent me them to clone. After opening, these I refrigerate.
