Yes it is possible, people use Cardboard for test benches as they dont short out the Mobo.
In fact, I think there is a project log somewhere on Toms where somebody did this as a testbench. The Mobo was attached to a peice of thick cardboard much like it would be to a normal Case, and the cardboard was upright. They were using a huge external Rad and watercooling. It looked awesome, there were decals and drawings allover, and the cardboard ended up looking immense with the designs.
Though is there a chance of the components catching on fire when I turn it on?
Yes, a minimal risk of fire hazard. Components that get hot will need to be cooled, fans being the best option. I've actually seen a number of cardboard cases, I recall an old issue of CPU mag that had one that a guy built on his wall.
Over this last summer my buddy built one mainly to house some fans. Used it to cool his work station. But really, one can build one fairly safely. Just remember you still have to worry about things like air flow and stuff.
I don't know if that's the one made with a VIA C7 or not. Basically you can do it with lower power components like an AMD C-50 or something. If you want to use midrange components it's a bit more difficult and you have to make sure that no single point can get hot enough to start burning. I definitely would do it with low power parts, but I definitely would not do it with mid range, let alone high end, parts.
Not too worry about fire. The autoignition point of paper is actually quite high (at leaset not the sort of temperature you will see in a working computer case). But it is flameable, so keep ignition source out (spark caused by short circut can be a hazard). I think you are sensible enough to not make spark with your components right amuffin?