Detective Burke and his partner, Detective Garrity, investigated the murders of three Wayne Enterprises executives who were killed on a subway train. They suspected Arthur Fleck of the crime, and attempted to apprehend him. After catching up to him, they chased him onto a train full of clown protestors. During the confrontation, a bystander is accidentally shot and a riot ensues allowing Arthur to escape.
Possible references to character origin
While Garrity may not be based on any detective from DC comics, Detective Burke could be based on the character of the same name from the DC comics "Gotham Central" series (known as Detective "Tommy" Burke in the series). in this series, Burke is a detective in the Major Crimes Unit, based in a 2000-2005 Gotham City.