I just had this idea… I think it would be really cool, and it turns out I’m not the only one. Loads of images on Google.
I wonder how amplified the danger of fire is? And I wonder if it would always smell like burning wood in the room it was in?
Also, this one is kind of ugly… I’m sure anything that one of us designed would come out much more aesthetically pleasing.
Not high on my list, but certainly a possibility for me.
You could also do one in acrylic 
I used to have an all acrylic case, it was pretty neat.
My husband just built a new PC and I plan to make a laser cut side panel once the GF comes in.
This one is quite pedestrian but you could go full steam or diesel punk with wood and make it very pleasing to someone that is into the look.
One of the reasons most PC cases are metal is for EMF protection, both incoming and outgoing. If you look in a laptop, the plastic parts usually have a thin metal cover on their inside faces. If you elect to do a case like this out of wood, you may see some unexplained undesired behavior.
I have also built an acrylic case but as far as wood goes you are not at risk of fire figure 3btu per watt. Worst that happens is the hot dry air dry’s out the wood and causes cracking but your pretty safe. I would suggest poly inside and out as dust clean up on would isnt that fun if it’s not sealed and you don’t want to have to use a damp rag to get it all off. So the poly coat you can wipe away dry and gives you a nice clean shine
And add LEDs of course with cool laser etched designs to be edge lit.
Obviously 
Some nice inlay/marquetry would be interesting on the side of a case like this.
When I look at that case I just see airflow problems.
Yeah, it needs a plethora of vents.
Definitely needs more ventilation.
You’d probably need to start with one of those older computer cases where the sides and top came off as one piece. All of my recent PC builds use cases where the top is permanently affixed to the chassis (though you could still skin it I guess.) Doing the chassis and case as all wood or acrylic, while very cool, would take a lot of tedious design work.