David Mitchell attended Kutztown State Teachers College, where he was first introduced to theater. He spent two years in the Army, after which he studied theater design as a graduate student at Boston University. There his teacher was Horace Armistead, a designer for theater, opera and ballet and an early Tony Award winner for set design.

In New York, Mitchell worked as an assistant to Ming Cho Lee for several years and, through him, began a longtime association with Joseph Papp, designing sets both for Shakespeare in the Park and the Public Theater. At the New York City Ballet, he designed works for Jerome Robbins and George Balanchine, as well as "Sleeping Beauty." His opera designs included works at the New York City Opera ("Mephistophele") and other U.S. opera houses as well as Deutsche Oper in Berlin ("Aida"). As a production designer for film, Mitchell's credits include "Rich Kids," the Paul Simon picture "One Trick Pony" and "My Dinner with Andre."