If you decide that you have a sharp edge at the hole through the tuner shaft, the round needle file, or the alternative, a small toolmaker's whetstone, moved across (perpendicular) to the shaft across the top of the hole is the standard method, but "de-burring" a small hole through a round shaft to get a really smooth edge can be sort of touchy. It's possible for the file or stone, to push a "hanging burr" that will stay sharp no matter how much you shove it around. You may not be able to see one of these "hangers" sharp enough to cut into your string with less than about 10x magnification.

For a final "polish" on the edge of the hole, just use a short piece of one of your old wound strings, pass it through the hole and work back and forth while you pull tension against the side of the hole where the sharp edge is. It won't cut anything very fast, unless you use an enormous amount of tension on the string, but just sliding it along the edge will get rid of any "hanging edges" on the hole.

The summer that I worked at the de-burr bench at a local airplane factory, I kept a piece of leather boot lace handy, with a cup of valve-grinding (fine) compound I could wipe onto the lace, just for that kind of hole.

John