No, Hershey syrup does not need to be refrigerated after opening. It can be stored at room temperature.
Yes, simple syrup should be refrigerated after opening to maintain its freshness and prevent spoilage.
Yes, pickles should be refrigerated after opening to maintain their quality and prevent spoilage.
no
Yes you will need to refrigerate after opening, but it doesn't need to be refrigerated if still sealed.
Raw maple syrup - such as you would buy at a country stand where they make it on the property - should be refrigerated, yes. Store bought maple syrup does not need to be refrigerated, in fact one well-known brand states on the label: "Refrigeration not necessary". Unless you use your syrup within two or three weeks after opening, it helps to refrigerate it. I've had open syrup develop a type of mold after a few weeks even though it was labeled not to require refrigeration.
Yes, Frank's sauce should be refrigerated after opening to maintain its quality and prevent spoilage.
Peanut butter does not need to be refrigerated, but jelly should be refrigerated after opening to maintain freshness and prevent spoilage.
Yes. It really should have been refrigerated before opening, too.
Yes, it is recommended to refrigerate simple syrup after opening it to maintain its freshness and prevent spoilage.
No, it doesn't need to be in the fridge. And being at a colder temperature will make it somewhat harder to pour out of the bottle. It's fine to keep it in the pantry and it's also great if you warm it up first.
Peanut butter does not need to be refrigerated, but jelly should be refrigerated after opening to maintain freshness and prevent spoilage.