The heat shield between the converter and the ground is there to prevent an accidental fire in the event you park on top of tall dry grass with a hot engine. You don't need it. I had the same experience as cousincletus with an Accord years ago. I took it off after it started rattling, and never had a problem again.
The reason that these heat shields don't last is because they are made out of aluminized carbon steel, and not stainless like the rest of the exhaust system. The heat shields are not part of the emissions system, so the manufacturer doesn't have to warantee them to last.
Heat shields located between the converter and the floorpan are a little different. They prevent the floor from getting too hot. Some converters that are close coupled (attached directly to the manifold) have heat shields that protect certain areas from getting too hot, such as wiring or electronics. If your heat shield is shielding heat from some wiring, then I would think twice about just removing it. You might want to bite the bullet and get one from a dealer.
