The club said Burke had resigned for personal reasons.
Burke arrived at the club in summer 2012 to take up a youth coaching role but replaced Gareth Evans as manager, initially on a caretaker basis, in February.
Livingston sit bottom of the Scottish Championship with one point from four games following an injury-hit start to the season.
A club statement read: "Livingston Football Club today accepted the resignation of Richie Burke as manager of the club due to personal reasons following his return from a brief trip home to see his family in the United States.
"The club acted quickly to appoint former Hearts manager John McGlynn, who will take up the reins with immediate effect."
Former Raith Rovers boss McGlynn, who was sacked by Hearts on the same day Burke took over at Livi, told STV: "It's a great opportunity for me at a very good football club and the decision to join Livingston was a no-brainer.
"We are an ambitious club and we want to reverse where we are in the table. I'm looking forward to that challenge.
"The offer did come out of the blue but it's fantastic to get back into football."
McGlynn was interviewed for the role before Burke was appointed on a permanent basis in March.
- Sports & Recreation
- Livingston Football Club