Why would the light that illuminates the dashboard and the heater panel be out after installing a radio in a 1996 Corolla when everything else works fine?

Check to see if you have a loose wire hangin out behind the radio... Typically this is caused by the dimmer wire for the dash lights.. you also may have hooked that wire to the wrong wire on your radio harness.. Good luck I would say its a fuse, or you crossed a wire, and your bulb burnt out. Hopfully, its the first one!