Born in Windsor, Ontario, Burke starred with St. Michael's in the MTHL before joining the OHL's Toronto Marlboros. After winning 25 games for the Marlies in 1984-85, he was selected 24th overall by the New Jersey Devils at the Entry Draft. Burke was returned to junior and toiled admirably on a weak defensive team in 1985-86.
After graduating from junior, the talented netminder opted to join the Canadian National Team. He backstopped Canada to 46 victories over two seasons then shared the goalkeeping responsibilities with Andy Moog at the 1988 Calgary Olympics. Following Canada's fourth place finish, Burke joined the Devils and caught fire at the right time. He posted a 10-1 record down the stretch and helped New Jersey make the playoffs on the last night of the regular season. In the playoffs, he was superb as the young club made it all the way to the seventh game of the semi-finals.
Burke was a workhorse during his first two full seasons before his play began to slip in 1990-91. The next year he was involved in a bitter contract dispute with the club and opted to rejoin the National Team. He posted an 18-6-4 record that year before backstopping the squad to a silver medal at the 1992 Albertville Olympics.
Prior to the start of the 1992-93 season, Burke was sent to the Hartford Whalers in a deal that also involved young forward Bobby Holik. He was brilliant at times for his new club but the franchise struggled with consistency. Burke remained with the club when it relocated to Carolina in 1997. The 1997-98 season saw the veteran netminder traded to the Vancouver Canucks then join the Philadelphia Flyers at the March trade deadline.
After the Flyers fizzled in the post-season, Burke signed as a free agent with the Florida Panthers. He played some of the best hockey of his career in 1998-99 with three shutouts, a 2.66 goals against average and selection to participate in the NHL All-Star Game. During the early stages of the following season he was on the move again when the Phoenix Coyotes obtained his services. Burke played well the rest of the year then sparkled in 2000-01 with a 2.27 goals against mark. He backstopped Phoenix to 25 wins and recorded four shutouts and entered the 2001-02 season as the undisputed first stringer for the club and was instrumental in leading the Coyotes to the 2002 playoffs.
Injuries plagued Burke's 2002-03 season as the Coyotes and Burke were never able to get things going. He redeemed the year of frustration by leading Team Canada to gold at the 2003 World Championship. Coming off a strong performance at the World Championships, Burke looked to rebound in 2003-04. In 32 games with the Coyotes, Burke had a 10-15-5 record but lost his number one status after the emergence of Brian Boucher who went on to set an NHL record with five straight shutouts. Burke was traded to the Philadelphia Flyers where he would finish the season before signing as a free agent with the Tampa Bay Lightning in the summer of 2005.
In 2005-06 Burke would backstop the Lightning for 35 games, while earning a respectable 2.22 GAA. The following season, Burke would be placed on waivers by the club, but would not be claimed. Burke would go on to compete in 10 games with the Lightning in the 2006-07 regular season before being sent to the club's AHL affiliate in Springfield. The veteran netminder struggled with the AHL club and as a result was placed on waivers once again.
On January 18, 2007 the Los Angeles Kings claimed Burke off waivers. He went on to post a 6-10 record on a struggling Kings team and announced his retirement in the offseason.
Aside from his Olympic experiences and the 2003 World Championships, Burke represented his homeland at the 1986 World Juniors and four other times at the World Championships (1987, 1989, 1991 and 1997)
