If you can somehow get it to start. See what it says in System Profiler under Diagnostics.
I actually just got done working on an MDD, that I think the Logic Board is failing. It would freeze randomly, so originally I thought there was bad CPU paste... I replaced it with fresh paste (Arctic Silver Ceramique 2) that I got from Radio Shack. It was a little more affordable than the Arctic Silver 5.
Anyhow, if I close the door on the side even slightly it shuts off. It also fails start up because of the RAM slots failing. I got three working so far.
Check out the RAM situation, as that caused the G4 not to pass startup tests (POST) and fail to chime. I think this G4's Logic Board is done though.