Cutting it is usually done as part of a ceremonial burning, especially where you have multiple people who each want a turn putting a piece on the fire. Not mandatory.

You can burn it in any fire, but traditionally you should be silent and respectful while it is burning.

The whole point of burning it is to not throw it away in a trash can and have it rotting in some garbage pile somewhere. So you burn it. If you want a ceremony, plenty of options, but if you want you can throw it in any campfire or home fireplace. Just ensure it's completely burned.