what would happen if I don't put a finish on my new computer desk?
Nothing disastrous will happen if you don't finish your desk top.  I see lots of old unfinished work bench tops and shelves at work, they are not pretty like old furniture but they're still working fine.  

It will be harder to keep clean - the finish on wood keeps dirt, scuffs, marks from sinking into the wood and makes them easier to wipe up.  Finishes also protect from water and sunlight, but that's more of an issue outdoors.  

There are all kinds of ways to finish wood, some are very simple and easy, no tools or oils required, and some don't involve any unpleasant fumes.  You could just wipe it down with an oil - some of these have very little odor, some even have a nice citrus smell.  If you want to seal it, paste wax is inexpensive, no odor, just takes a little elbow grease to apply.