Although a dead battery won't cause the laptop to overheat, it will cause it to shutdown unexpectedly.

Make sure battery connection is secure.

Make sure the Adapter is properly plugged into the wall and laptop.

Dell batteries come with a battery life indicator located on the bottom of the laptop. Press and release the test button. Each lights represent a 20% charge. If no lights are present, replacing the battery may be necessary.

Dell Latitude D820 Battery Replacement