I have two bottles of ketchup, one says refrigerate after opening, the other does not. The mayo says to refrigerate after opening. Peanut butter usually doesn't require refrigeration, unless it's all-natural (i.e., the kind you have to stir). And the bottle of maple syrup that I have also says to refrigerate after opening.

Things you should never store in the refrigerator:

Honey: it will crystalize faster
Bananas: the cold will stop ripening & make the peel turn dark
Coffee: will lose its flavor and pick up bad odors
Onions: will turn soft
Tomatoes: will stop ripening and turn mealy
Bread: You can freeze bread, but putting it in the fridge make it go stale faster

(You can put bananas in the fridge, but you have to let them ripen at room temperature first. Once they reach your desired level of ripeness, THEN you can refrigerate them and they will stay at that level longer than at room temp. But if you stick green bananas in the fridge, the skin might turn brown but it will be unripe on the inside.)