A three-dimensional (3D) metal object manufacturing apparatus is equipped with an orifice cleaning system that removes metal drops that have adhered to a plate, an orifice in the plate, and a nozzle ejecting melted metal drops through the orifice during object forming operations. The orifice cleaning system includes an orifice cleaning tool that consists essentially of a soft carbon material, such as graphite. The orifice cleaning tool is configured with a handle that is gripped by an articulated arm to move the orifice cleaning tool against the plate, the orifice, and a portion of the nozzle at the orifice.