Vincent Kompany won four Premier League titles at Manchester City
City legend Kompany, who played for the club from 2008 to 2019, has guided the Clarets to the top of the Championship in his first season in charge.
League Two Grimsby - the lowest-ranked side left in the cup - go to Brighton & Hove Albion and Manchester United host Fulham in an all-Premier League tie.
Sheffield United play Championship rivals Blackburn in the final match-up.
The Blades stunned Tottenham Hotspur - who are fourth in the Premier League - on Wednesday while Blackburn upset Leicester City in their fifth-round fixture.
Grimsby produced the result of the round by winning 2-1 at Premier League bottom club Southampton to reach the quarter-finals of the FA Cup for the first time since 1939.
The last-eight ties will be played from 17 to 19 March.
Manchester City v Burnley
Manchester United v Fulham
Brighton & Hove Albion v Grimsby Town
Sheffield United v Blackburn Rovers
-- Courtesy of BBC Sport