I think this idea could work, and I don't see why an experiment couldn't be done on Github--after all, it'd all be just text files.

People could fork, say, a short story, rewrite parts, and submit a pull request; the original author would maintain control in this case. If the original author didn't like the change but someone else did, that person could fork the forked version (the one containing the change), add additional changes, then submit that as a pull request, etc.

It could get messy, but it also sounds like it could be really interesting, with the end result being a tree of competing stories.

Maybe somebody could submit a short story, publicize its availability, and see what happens.