At about 1143 mountain time, a Beech BE-76 overran runway 21R after aborting a touch-and-go takeoff at Love Field Airport. The flight instructor and private pilot were not injured. The flight had been making touch-and-go landings, and on the third one the students door popped open. He aborted the takeoff but was unable to stop the airplane on the runway. Both pilots said the airplane was at takeoff airspeed when the door opened.