On my 1990 Honda, it's falling apart. I'd like to just rip it off; will that cause any problems? More specifically: will the car pass New York state inspection with a missing cover?
I can't tell you if it'd fail inspection or not, but there's a good reason for its existence.
It's basically a heat shield that protects the body of the car and the carpet/sound deadening in the car from the heat of the cat. Catalytic converters can get very hot (in the worst case, glowing hot) and without the heat shield that can cause problems or at least discomfort.
I'd try to see if you can find a replacement at a junk yard or on ebay from somewhere less corrosion-prone.