Originally Posted by Dan Stokes

Another option I've used but can't recommend:

With an oxy-acetylene cutting torch you can light the torch and point the flame DIRECTLY down the hole that the tap is in. Once the tap is nice and red or even hotter you hit the oxygen lever and keep your face out of the way. This will blow the now-molten tap out of the hole. It's dangerous and will screw up if the heat isn't applied directly in line with the broken tap (the hole will easily veer off). When I've done this I've subsequently oversized the hole and either tapped oversize and installed a bigger screw or Helicoiled the now bigger hole. Ugly but it can work. This is sort of a poor man's EDM.

Dan