If the screw is obscured by furniture (door handle or hinge plate), drill a hole at the side of the old screw to release the pressure of surrounding wood. The old screw should remove easily with pliers or similar.
Take a length of dowelling too long for the job and the same diameter as the drill used and apply a smear of wood glue to the length of it. Do not pour glue into the hole, trapped air can become compressed when the dowell is tapped in and may damage the door at the other side if the hole is deep and close to the other face.
Tap the dowel into the drill hole and leave the glue to dry. Using a sharp chisel, tap around the dowel resting the bevelled edge against the door. Snap off the excess dowel and tidy up the remainder and any glue that may have been pushed up out of the hole.
Before you try to use a screw in the original hole, do as suggested and pre-drill the hole to the correct depth, go gently with a power drill having marked the depth you need with electrical tape on the bit or use a hand drill for more control. Or use a steel screw warmed and coated in wax from a candle before using the brass one if used, again as previously suggested.
If the screw hole isn't covered over, there is sufficient screw left sticking out and you need to preserve the finish of the surrounding wood. Mole grips or similar can be adapted by filing a v in the middle of each jaw, before clamping the grips tightly to the screw, take some HDPE from a carton (recycle number 2) big enough to cover the immediate area, make a hole in the middle and slip it over the screw. Say a little prayer and try to turn the grips counter clockwise until the screw loosens at which point remove the grips and finish removing the screw by hand.
Good luck
Tinker
March 2009