Batteries it depends on the chemistry.

NiCad/NiMH and some other older battery tech it can prolong shelflife, especially if you're in a hot/humid climate. Lithium batteries it doesn't really do anything.

Some soy sauce recommends refrigeration on the bottle. Some don't. I don't know why.

I like my ketchup refrigerated, not because it has to be, but because I like the contrast between cold ketchup and hot food.

If you use "natural" or "organic" peanut butter (read, ingredients: peanuts) you'll want to refrigerate it. It helps keep the oil from separating. It'll be the consistency of jiff or other sugared brands for most of the jar. I usually take mine out of the fridge when you get towards the bottom so it doesn't get too hard.

Refrigerating bananas pretty much stops them ripening, so if you have some fully yellow bananas you can pop then in the fridge and it will stop them from over ripening for a few days. The peel will still go brown but the flesh remains as it was when you put them in. You definitely shouldn't put green bananas in the fridge, but with yellows it buys you some time.