"Off" indicates an open circuit for household applicances, lights, etc. Maybe the original fire alarms normally had closed circuits, and began to sing when their circuit was opened

Good point. I don't know if it's the same for everyone, but in my house there are little magnets on the tops of the doors and windows which make the alarm sound if they are separated from their mates in the door/window frames, thus 'opening' the magnetic circuit [not-a-very-good-grasp-of-physics-e]