The aircraft impacted trees and a hillside shortly after a night takeoff from runway 33 while the CFI was flying the aircraft from the right seat. The CFI reported that he intended to fly the pattern to demonstrate a landing to his student. During initial climb, he lost sight of the terrain ahead of the aircraft and subsequently flew a heading of 300 degrees using the DG. The aircraft subsequently hit trees and came to rest nose-down on a hill about 1/2 mile due north of the airport. The CFI reported that he knew the DG had a tendency to precess excessively and that he had reset it prior to takeoff. Post-crash testing of the DG revealed that it failed to operate within manufacturer specifications. Total reported operating time on the unit since new was 1460 hours. The airport was observed to be equipped with lights along only the first 1000 feet of the runway. The CFI reported that it was accepted local practice to depart only on runway 15 at night due to the lack of visual cues in the hilly terrain to the north of the airport.