The move comes after the 20-year-old signed a new contract with the Royals, which will keep him at the Madejski Stadium until the summer of 2017.
Tanner made his debut for Reading in August and has scored one goal in five appearances for the Berkshire club.
Reading, who are 18th in the Championship, have the option of recalling Tanner after a month.
AFC Wimbledon manager Neal Ardley said their scouting reports showed Tanner was "one of the best young talents around".
He told the club website: "With the qualities and talent that Craig has, we believe he could be an exciting player who can cope with the demands at this level."
League Two side AFC Wimbledon have signed Reading striker Craig Tanner on loan until the end of the season.