Following his playing career, Burkhart developed as a well-regarded umpire. He worked in three World Series (1962, 1964 and 1970), serving as crew chief in 1970, and in the 1972 National League Championship Series. He umpired in four All-Star Games (second 1959 game, second 1962 game, 1967 and 1973), and also umpired in no-hitters on consecutive days in for Gaylord Perry (September 17) and Ray Washburn (September 18). On May 11, he was umpiring at second base when Sandy Koufax pitched his second no-hitter, and on June 21, he was at third base when Jim Bunning of the Philadelphia Phillies pitched a 6–0 perfect game against the New York Mets. Burkhart officiated in eight no-hitters overall, then one short of the record for NL umpires, but did not work behind the plate for any of them. He was, however, behind the plate on July 12, when brothers Tommie and Hank Aaron both hit home runs in the ninth inning to propel the Milwaukee Braves to an 8–6 win, with Hank's grand slam winning the contest.