Why would a 1985 Chevy 305 keep burning out ignition modules?

You should check the pick-up coil, it can short out the modules. I have ran into this problem before and was a the end of my rope. I found a pamphlet at the auto parts store and read through it and came to the conclusion that the pick-up coil was shot, replaced it and never had to put another module in.

Probably lack of die electric grease under the mounting surface of the module. This special grease keeps the module cool & lack of it could cause the module to overheat & burn out.