One interpretation (? - could be authentic rather than a sound effect) of ringing during steel driving can be heard in the John Henry segment of the Disney movie "Tall Tale" (1995). There each blow of John Henry's hammer on the steel results in a prolonged "boooong!" This kind of sound would have to be emitted by the steel drill. Indeed, I've now found other references to the physics of the ringing sound of the steel drill.

Basically, compression waves move longitudinally up and down the drill. As they pass a given spot, the steel deforms (very slightly) laterally, displacing air and creating sound waves. This type of motion is said to be little affected by hands on the drill or mud in the hole, that is, these things don't damp the sound very much.

The ringing of axes in a forest is a matter, I think, of echoes. The ringing of a steel drill in a tunnel would echo, too, but I don't think that the basic ringing sound has anything to do with echoes.