The Rays selected him in the third round, with the 96th overall selection, of the 2014 MLB draft. He signed for an above slot $897,500 signing bonus. He made his professional debut in 2014 for the Gulf Coast Rays, going 0–3 with a 10.80 ERA in innings pitched. Burke spent 2015 with the Princeton Rays where he went 4–2 with a 3.42 ERA in 11 starts, and 2016 with the Hudson Valley Renegades where he pitched to a 3–3 record and a 3.39 ERA in 13 starts. Burke began the 2017 season with the Bowling Green Hot Rods, and was promoted to the Charlotte Stone Crabs during the season. In 23 starts between the two teams, he went 11–6 with a 2.99 ERA. In 2018, he pitched for both Charlotte and the Montgomery Biscuits, compiling a combined 9–6 record and 3.08 ERA 25 games. The Rays added Burke to their 40-man roster after the 2018 season.
Texas Rangers
On December 21, 2018, the Texas Rangers acquired Burke from the Rays as part of a three team deal in which the Rangers also acquired Kyle Bird, Yoel Espinal, Eli White, and $750,000 of international signing bonus pool space, the Rays acquired Emilio Pagan, Rollie Lacy and a competitive balance pick in the 2019 MLB draft, and the Oakland Athletics acquired Jurickson Profar. In 2019, Burke was optioned to the Frisco RoughRiders of the Double-ATexas League to open the season. Burke was placed on the injured list on April 23 due to shoulder fatigue and blister issues. He made one start each for the AZL Rangers and the Hickory Crawdads, before returning to Frisco on June 26. With Frisco, he went 3–5 with a 3.18 ERA in 45 innings. On August 6, Burke was promoted to the Nashville Sounds of the Triple-APacific Coast League. He made two starts for Nashville, allowing 7 runs over 8 innings. On August 20, 2019, the Rangers promoted Burke to the major leagues. He made his major league debut that night versus the Los Angeles Angels, recording four strikeouts over six scoreless innings. With Texas in 2019, Burke went 0–2 with a 7.43 ERA over innings.
Personal life
Burke has suffered from somnambulism his entire life. On multiple occasions while in the minor leagues, Burke was witnessed by different roommates waking up suddenly during the night and acting out in various ways.