Is there a theory for what "effect" would be
     produced by a theoretical object going faster than 
     light?

There really isn't an effect. The speed of light (in a vacuum) is just the speed of causality. The fabric of space-time will bend around you to ensure you can never get there. Objects without mass move at the speed of light, and no other speed.

Now if you don't mean in a vacuum but in a medium like water, or air, or ice, etc.

That is https://en.wikipedia.org/wiki/Cherenkov_radiation