Does a PC have to be mounted in metal case
Solution 1:
A computer case does not need to be made of metal. The reason most cases are metal is because they conduct heat (helps disposing of internal heat a bit better if there are few fans inside), and more importantly, its cheaper to produce and the end product is less heavy. A computer case is not being used as ground, simply because the case itself does not touch the floor. They usually have rubber feet.
The case may be connected to the grounds of the motherboard, but that is not necessary, and it is mostly done in case the computer is not connected to a grounded wall-outlet.
That said, I've seen many people build computers from various materials, wood included but not limited to. Test pc's often have no case at all, but the best example is the desk pc, a pc build directly into a desk. These desk pc's are often made of wood.
Solution 2:
To pass the requirements of FCC part 15 class b most computers must be in some sort of conductive case. You will notice that most metal cases don't have any slots or holes larger than about half a centimeter, this has to to with antenna theory.
If you are building the projects for yourself, you do not have to comply to FCC part 15, but if you intend to sell them that is a different story. If you want to build a case out of wood and sell the resultant product (not as a kit, there is a whole weird legal thing about kits) you will almost certainly have to coat the inside with some sort of conductor.
Solution 3:
I've been through the guts of hundreds of computers, both desktops and laptops, and have re/built a number of them, often working on them with only a frame or even no case at all-- just parts on a bench. I've never seen any difference. I also play electric guitar, sensitive to radio and electronic interference, and can attest a pc case makes no difference.
My favorite mini-server I built for low power-consumption from the guts of an old HP g-series laptop, bolted to a sheet of plexi-glass. All the foil shielding makes no difference. It's run for years without any errors caused by electronic interference, transmitters, cosmic rays, or solar flares-- though that last bit did measurably affect wifi communications between other devices and the AP's, but those were on another network segment and physically seperated by a router and 5 to 100 feet of space, so obvi would not have been in the case anyway.
I actually tested this with various transmitters and power-supplies brought in close proximity. No perceptable difference, and nothing in the error logs to suggest it was affected. A guitar pickup likewise is affected the same by PC's with or without a case. Kind of a cool, practical test for outgoing interference as you can actually hear it. Also necessary if your a guitar player. Grounding the PC also seems to make no difference to the PC in terms of function, though possibly in terms of safety in extreme cases, but you don't need a box for that.
If it is a concern, and you like wood cases, you can always line it with foil, or foil tape. A sheet of Reynold's behind your AP rounded a bit like a dish works wonders for reflecting or focusing your wifi signal, just like the reflector in a mini-maglite. You can test it with a wifi analyzer app on your phone, so it will definitely do a lot to keep interference in or out of your box. But like I said, I have tested it and see no practical difference.
For an easy real example of interference, try having a conversation, listening to Pandora, downloading something etc. on your cell-phone within 3 to 5 feet of your average microwave oven. If it's not one of those cheap dinky one's, you will notice significant chop, or even total signal jamming. If that's considered safe, I think I'm ok without a metal box. My naked server sat on a shelf on my desk less than two feet from my skull. I've never noticed any difference there either. More worried about my laptop in my, well, lap.
Now... where did I put my tinfoil hat? ;)
Solution 4:
I don't imagine you were thinking about Raspberry Pis (or other SBCs), but I'd estimate that 80% of all Raspberry Pis ever sold are either in plastic cases, the cardboard box it was posted in, or in no case at all.
A metal case isn't a hard-requirement, although can offer some benefits if you want to do some EMI shielding. You can do the same shielding with some metal that's not integral to the case though, if you need want the shielding without the case.