Treat a two-stroke ice auger motor just like you would a two-stroke outboard, chainsaw, or whatever that you are putting in storage--the first steps also apply to a four-stroke, with one exception I noted below:
Fill the gas tank, and put in the recommended amount of fuel stabilizer.
Run the motor for about five minutes to be sure the stabilized fuel gets in all the lines, carb, and chamber.
Cut the fuel off and let the motor run out, as you don't want too much gas-air mix stuck in the carb for months--this can be hard on the seals.
THIS NEXT PART ONLY APPLIES TO TWO-STROKES: remove the spark plug(s), and spray the combustion chamber with fogging oil. Also spray some fogging oil in the carb. Pull the start cord a couple times to coat the cylinder walls with fogging oil. DO NOT USE FOGGING OIL ON FOUR-STROKE MOTORS--IT WILL GUM THEM UP.
Put your motor away for the year. It should start right up next fall or winter. The fuel stabilizer keeps the gas from breaking down in to turpentine, which is why you want to make sure mixed fuel gets all through the motor. A clean tank of gas does no good if there are pockets of turpentine in the carb, fuel lines, and chamber.
In a two-stroke motor, fogging oil protects the carb and cylinder walls from moisture and corrosion that can build up while it sits. Again, DO NOT use fogging oil on a four-stroke.