Butter? I think that's a no-brainer. Of course, we do. Nothing quite as disgusting as rancid butter.
Ketchup? Yes - Although it doesn't spoil easily because it is a very acidic product, it does keep longer in the fridge.
Mustard? Yes - It will not spoil outside of the fridge but it loses its flavour much more quickly.
Syrup? No. We use real maple syrup too (being from the land of maple syrup). If you refrigerate it, it will preserve it's freshness. However, unrefrigerating it will not cause it to go bad.
Soy Sauce? You don't have to refrigerate it even if the package says to. We don't.
Vinegar? Why in the world would you ever refrigerate plain old vinegar? NO.
Peanut Butter? Depends on the type.
Oh, to the person whose child likes to drink vinegar, just a little warning. That could be incredibly harmful to their insides, if they are drinking it in any great quantity. It can harm the stomach wall and create ulcers.