Not necessarily, unless your components are constantly shorting out and sparking everywhere 24/7, wood or even paper would totally be fine to make a PC case out of. Ignition temp of wood is something like 500F, and paper is like 400F. If your PC is producing those kinds of temps, you probably have some bigger problems than having a wood or paper case
However, this does depend on your definition of "case" here. You can't just put a box together with no airflow if your components use air cooling of any kind, you'll need to make vents so you can get some proper airflow throughout so things don't overheat. As long as done properly, pretty much any sturdy material could be used as a case probably. You probably won't want to use something like giant magnets or something, but otherwise eh, sky's the limit.
I would avoid materials that tend to trap heat in easily, carpeting as mentioned above probably isn't a good choice for multiple reasons; heat for one, as it's more likely to hold in heat that you're trying to expel from the system, which may cause components to overheat and shut down, and then obviously carpet itself isn't sturdy enough to actually hold components unless you're bolting the carpet itself to some kind of fibreboard or something
But engineer it properly means be aware of your airflow/cooling and play to it and "don't twist the motherboard". For a higher performance rig like some build for games the cooling can be worth considering but for your bog standard air cooled affair with a sensibly priced GPU is it nothing too drastic.
Not like it is a house that is going to fall over or other sort of thing likely to be subject to serious forces.
Pretty much everything can be used as long as it doesn't short circuit anything and can handle the temperatures of your system while providing adequate air flow. I've even seen a book PC working properly (and by book PC I mean literally a thick book with the pages cut out leaving only the surrounding frame of them all around and having the components put in there).
You can make a PC case out of a giant plastic bucket and fill it with mineral oil. Submerged PCs are cool... at least until you need to take it back out to replace something - cleaning them is a massive chore.