High salt/vinegar content condiments are perfectly fine at room temp for a weeks to months in dry to mostly dry moderate temp climates. That is why air conditioned restaurants which have consistent temps and low humidity leave them out on the tables.
The label is there so someone in Florida doesn't have it go bad in a couple months on their counter. Plus refrigeration extends the time it can go without spoiling, which is great for condiments that are rarely used.
…the reason jelly/jam/preserves are canned is because they are not shelf stable otherwise. I just threw out a jar because it molded in the fridge…
Peanut butter is shelf stable, but we usually get the stuff that’s just peanuts and salt, so it separates at room temp.
Mustard, ketchup, & soy/fish sauce… sometimes it’s just convenient to keep most of my bottles and jars together in the fridge door.
I’m hypersensitive to rancid oil. Also the healthy parts of olive oil & fish oil degrade with time, heat, sun and oxygen exposure. The fridge slows this down. That said, I keep my cooking oil under the counter.
If it's in a sealed plastic bag it doesn't go stale until long after it would have molded on the counter. I refrigerate mine because I buy Costco sized sliced bread and it takes me 2 months to go through it. If you toast your bread, the staleness is unnoticeable
A lot of these things only need to be refrigerated to preserve flavor, not to stop spoilage. If you go through a bottle of ketchup in 3 months there is little benefit to refrigerating it, if it takes 3 years for you to finish it, it should probably stay in the fridge.
Some peanut butter brands require refrigeration to prevent mould. Others recommend it because it stops the oils from separating. Brands like Kraft don't require any refrigeration at all
Refrigerating oil will stop it from going rancid, but I've only ever needed to do this with used deep frier oil
Same for honey, as cold accelerate the crystallization process.
Peanut butter is basically oil already, but putting it in the fridge might help keeping it less oily. I eat organic 100% peanut butter and it is often oily when I open it. I think that’s why some have palm oil in it.
Soy sauce should be salty enough to store out the fridge but I prefer to keep it in the fridge for some reason.
Oyster sauce contains sea food, so straight in the fridge!
I used to buy a lot of """"natural"""" peanut butter. The kind in glass jars that separates after a while, so you have to stir the jar every time you use it. After a while, I started keeping it in the refrigerator because that stopped it from separating at all. Just stir once when opening the jar for the first time, then into the refrigerator it goes, and it never needs stirring again.
Honey depends on the quality. Real honey will basically never turn bad (they found containers with thousand year old still edible honey), but the cheap stuff is sometimes mixed with sugar syrup etc. and then it needs refrigeration.
The refrigeration is either to extend flavor or to prevent spoilage in hot and humid locations where mold can build on the parts of the container that dry out if it isn't used often.