To anyone else who is having this issue...
I had the same problem in Arc 10.2- I would click "Update Georeferencing", the software would "think" for a few seconds, act as if it completed the task, but then the raster would reload to it's original position (in my case at 0,0). There was no obvious update to the raster file, and all of the linked points in the georeferencing window would disappear, as if they had been applied and then cleared.
The specific setting that I changed to correct this was
Customize > ArcMap Options > Raster > Raster Dataset >
and Un-check "Use world file to define the coordinates of the raster"
As far as I can tell, that setting being checked prevented any georeferencing information being written to the file at all, i.e. not to a world file, the aux file, or the tiff itself.
I suspect that this is not the intended behavior, and is a hassle for me, since most of our company imagery uses world files for georeferencing.
Hope this helps someone down the road.