The main thing for a case is to keep your components safe... This helps if it ever gets bumped or knocked over...
Secondly without a case dust will accumulate much faster... This will cause over heating, and bad connections between stuff like your CPU heat sinc and cpu...
But, yes you can run without a case and it is very easy, many common designs are to drill holes the size of your MOBO under your desk and mount your mobo, then attach your CPU, GPU, and ram... Then drill more holes to mount your PSU and HD/SD.
I would recommend also mounting your fans near it to keep air flow, as a hot computer does not run well, but this is not a requirment as it is open to the environment so to speak... Many of the most state of the art cases right now have 1 or maybe 2 fans and are open to the environment (Granted they are water cooled though)... Where as 5 years ago the most common designs were sealed cases with over 7 fans.
Also if this is a computer bought off the shelf, the only fan you are likely to have is your heatsink. I would suggest buying a new case, I got a DIYPC brand for $30, just make sure your mobo will fit your case if you get a Atx or MINI style.
If you need help finding if the mobo will fit I can help you find out.
Lastly DIYPC comes with 2 fans most of the time, I would suggest buying a 3rd. Mount two in the front and one in the back near the CPU. This will create a positive pressure in the Case making dust less frequent.
Another thing that might help, you can enter your BIOS on startup and set your fan speed for operating temperature ranges on most PC bios'.
I like to make my Heatsink fan go 100% at 75c and above. Just to be cautious, I don't care much for the noise as my fan is very silent.