Here's the method I use.
Oil the piece with organic oil and rub in a curcullar motion with a smooth object. I have never done this while the piece is rotating, only while I hold it myself. The process takes time and patience. Do not press hard. Use a steady, even pressure. If you press too hard, you leave little whorls on the piece. Do not over-burnish. The burnished area will start to powder and flake off if you do.
The river rocks used for this process must be very smooth, ie tumbled through the river bed for a long time. They are hard to find. You can use rocks smoothed with a rock tumbler. Any lapidary shop usually has some for sale. Or, the easiest, a good quality stainless steel spoon back works great. Just be sure it is hard enough that it doesn't leave metal on the piece.
Take frequent rests, especially if you've never done this before. A dear friend of mine who's work had just been accepted by a major gallery developed carpel tunnel syndrome and had to quit making his beautiful burnished pots. I am fortunate to have one his first pots that I bought at a local art fair.
Fire no higher than cone 04. I have seen pit-fired, dung-fired, and sawdust-fired burnished pieces. Higher temp will destroy the burnish. Finished pieces are pourous and not considered food safe. However, they are beautiful decorative pieces. Vases make wonderful containers for dried folliage and flower arrangements.
Jan C.