When putting food that was just cooked in the freezer or refrigerator, does it need to be cooled first?

It does help to cool it before going in the fridge, but it's more about ambient heat than the food that is cooling. If you put a hot food in the fridge next to say, a carton of milk or a package of chicken, the ambient heat warms it. This can cause things to go bad, and potentially make you sick, so be weary of placing hot things in the fridge.