I am going to have tile put over the existing tile in a shower which should raise the floor about 1/8th inch. Is is possible to get an extension to insert into the existing drain to compensate for the tile? What does the part look like and is easy to install? I have done some plumbing work and can do it if it doesn't require special tools or special knowledge.

I'm not a tile man and I hope one of the other experts has a better grasp then I do but 1/8" of a inch isn't very much. I've seen drains that were recessed much more then 1/8th inch. The tile man tiled up to the drain and then ground ,sloped and polished the tile so it joined up with the drain lip. This is much easier then hassling with the drain.
However if you wish to raise the drain it screws into the flange type shower drain,(see image) . However you'll have to break the drain loose from them old tile before you can turn it. Good luck. Tom

Absolutely. The drain you have may even be adjustable, but there's no point in tearing up tile to get to it.

As speedball suggests is how we've done it a few times over the years.