Or when there's no oil for the drilling rig to pump? I just tried the oil drill for my first time (in previously playthroughs I used oilberries or oilsands ore instead), and I don't see a way to do that, even looking at the source for the drill (I haven't tried the advanced miner ii). It looks like one needs to use a pickaxe (or a tool that counts as one, such as a drill) to recover the mining pipes.
I might also want to retract the mining pipes if I want to move on to get a different kind of oil, or stop the advanced miner ii and move it somewhere else. I'd write up an issue for it, but I'm not sure what to suggest for signaling to the multiblock that I want to retract the mining pipes - with the IC2 miner, the Wiki says one removes the drill to tell it to start retracting the mining pipes, but the drilling rig and advancer miner ii don't have removable drill bits.