Burke's staff assignments included tours as an instructor and director of the Officer Department Electrical Engineering Division at Naval Nuclear Power School in Orlando, Florida; junior board member on the Pacific Fleet Nuclear Propulsion Examining Board; and as the Submarine Officer Community manager/Nuclear Officer Program manager in the office of the Director, Military Personnel Plans and Policy Division (N13).
Following command, Burke held assignments as senior member, Tactical Readiness Evaluation Team, on the staff of Commander, Submarine Force, United States Atlantic Fleet; as the deputy director for Operations, Strategy and Policy Directorate (J5), at United States Joint Forces Command; as the division director, Submarine/Nuclear Power Distribution (PERS-42) and Nuclear Propulsion Program Manager (N133), Director, Joint and Fleet Operations (N3/N5), United States Fleet Forces Command and most recently as deputy commander, United States Sixth Fleet, director of Operations (N3), United States Naval Forces Europe-Africa and commander, Submarine Group 8.
Burke was awarded an Honorary Officer of the Order of Australia on 24 August 2020 with the citation being "For distinguished service in strengthening the military alliance between Australia and the United States of America".[5]
The Naval Submarine League recognized Burke with the Jack Darby Award for Leadership in 2004. Burke also received the Vice Admiral James Bond Stockdale Award for Inspirational Leadership in 2005.