Prior to turning professional, Burroughs played for the Regina Pats and the Medicine Hat Tigers in the Western Hockey League.

He then established himself with the Valley West Hawks in the BC Hockey Major Midget League where he was also named an associate captain.

[7] During the 2013–14 season, Burroughs suffered a concussion as the result of a hit by Brandon Wheat Kings centre Jayce Hawryluk, who was suspended two games.

On January 5, 2015, Burroughs and Dryden Hunt were traded to the Medicine Hat Tigers in exchange for Connor Hobbs and future draft picks.

[12] While playing with the Medicine Hat Tigers in the 2015 WHL playoffs, Burroughs signed a three-year, entry-level contract with the Islanders on April 8, 2015.

[13] Burroughs contributed three points in ten games during the playoffs as the Medicine Hat Tigers lost to the Calgary Hitmen in the Conference Semi-finals.

[15] On March 6, 2016, Burroughs collided with a player from the Hartford Wolf Pack, causing him to miss a month of playing time.

On March 30, 2017, Burroughs was named the Sound Tigers AHL Man of the Year due to his work in the community and with charities.

[28] After attending the Avalanche training camp leading into the delayed 2020–21 season, Burroughs was placed on waivers and upon clearing was assigned to their AHL affiliate, the Colorado Eagles, on January 10, 2021.

[29] Burroughs played in his first NHL hockey game on April 5, 2021, in a 5–4 win against the Minnesota Wild at the Xcel Energy Center.

As a free agent after his successful stint with the Colorado Avalanche, Burroughs was signed to a two-year, two-way contract with the Vancouver Canucks on July 28, 2021.