In The Time Machine (Guy Pearce version), they manage to literally break up the Earth's moon with obviously disastrous results and I believe they were just building underground structures.
I am wondering if massive mining operations, shipping ore to Earth and thereby not just decreasing Luna's mass but also increasing that of Earth, could cause snowball affect on the orbit -- perhaps Luna would settle into a lower or higher orbit or maybe the decrease in Lunar mass is offset by increase in Earth's mass. But obviously some mass would be lost.
Anyway, given the immense masses involved, maybe we could not accidentally or even deliberately affect the orbit -- the latter I would not bet on, concerted effort involving a lot of nuclear explosions I bet could cause the orbit to decay but I see doing so pretty unlikely.