What doesn't belong in the refrigerator
Your refrigerator runs at one temperature. The produce you put in it came from dozens of different climates and wants dozens of different conditions. Most of the time the cold is the right answer anyway. Sometimes it's the thing doing the damage.
UC Davis's postharvest researchers make this point in the least dramatic way possible: fruits and vegetables can't be sorted into one storage condition. Their compatibility work splits produce into groups, with most leafy vegetables and temperate fruits wanting roughly 32 to 36°F, and a second group of cold-sensitive items sitting much warmer, around 45 to 50°F. A home refrigerator offers exactly one of those.
The tomato is the big one
This is the case worth knowing, because it's common and the damage is invisible until you eat the thing. Per UC Davis, tomatoes are chilling sensitive below 50°F if held longer than about two weeks, and at 41°F, which is roughly a normal refrigerator, the window is more like six to eight days. What goes wrong isn't rot. It's failure to ripen and develop full color and flavor, blotchy coloring, and premature softening.
That's the frustrating part. A chilled tomato doesn't look damaged. It looks like a tomato and tastes like wet cardboard, and there's no way to reverse it once it's happened. Counter, out of direct sun, until it's ripe. After that the fridge buys you a few days at some cost to flavor, which is a reasonable trade when the alternative is losing it entirely.
Basil, and only basil
Here's where the popular advice overcorrects. Most fresh herbs do want the cold. UC Davis is specific that among fresh culinary herbs, basil and shiso are the chilling sensitive ones. Basil in the fridge browns at the leaves and growing tip, its leaf veins bronze, the leaves lose their gloss, and cold storage markedly reduces its aroma.
So basil goes on the counter, stems in water, like a small unhappy bouquet. Parsley, cilantro, mint, and the rest belong in the refrigerator, and they'll thank you for it. Worth knowing about cilantro specifically: UC Davis notes its aroma falls off noticeably after about ten days even while it still looks sellable for as long as three weeks. Looking fine and being good are two different measurements.
Potatoes, with a correction
You have read that refrigerating potatoes ruins them. The research is more measured than the internet. UC Davis notes that holding potatoes near 32°F for a few weeks can produce a mahogany discoloration in the internal tissue of some varieties, and that chilling injury generally needs longer than that to develop. A potato that spent three days in your fridge is fine.
The more useful potato fact is about light, not cold: bright light, or even one to two weeks of dim light, triggers greening as chlorophyll develops in the tuber. Which argues for a closed cupboard rather than the decorative bowl on the counter.
What the cold is actually for
For most things, refrigeration is doing exactly what you want, slowing everything down. The exceptions are worth learning because they're specific and few: tomatoes, basil, cucumbers, and most tropical fruit would rather be warmer.
One adjacent thing, since it costs nothing. UC Davis's storage guidance treats ethylene as a real variable, recommending it be held under 1 ppm in commercial storage, and the cucumber guidance explicitly says not to store cucumbers alongside bananas, melons, or tomatoes. You can't measure ppm in your kitchen, but you can keep the bananas away from things you want to keep.
This is also why Savery treats storage as changing the speed of an item's clock rather than starting a new countdown. A tomato on the counter and a tomato in the fridge aren't on the same schedule, and for tomatoes specifically, the colder one isn't necessarily winning.