Between 1987 and 1988, he briefly had the lowest career earned run average for a relief pitcher,[1] and his career earned run average of 2.72 is lower than all Hall of Famer relief pitchers except for Mariano Rivera and Hoyt Wilhelm.

Drafted by the Pittsburgh Pirates in round two of the 1980 MLB draft, Burke was traded to the Yankees on December 22, 1982, and then after a year in New York's minor league system, was dealt to the Expos for outfielder Pat Rooney on December 20, 1983.

In an eight-season career, he posted a 49–33 record with a 2.72 ERA and 102 saves in 498 games pitched, all but two out of the bullpen.

In 1994 he authored the book Major League Dad: The Moving Story of an All-Star Pitcher Who Gave up Baseball for His Family.

In 1995, Burke appeared in the Geoff Moore & the Distance music video for the song "Home Run".