The Colorado Rockies selected him in the 25th round of the 2008 Major League Baseball draft, but he opted to attend college.

After sitting out a season due to the National Collegiate Athletic Association's rules on transferring between schools, the Toronto Blue Jays selected him in the 11th round of the 2011 Major League Baseball draft.

[5] Burns played for the Lansing Lugnuts of the Single–A Midwest League in 2012, appearing in 78 games and hitting .248 with 9 home runs and 37 RBI.

[4] He played in 128 games split between the two teams, and hit .288 with a career-high 15 home runs, 85 RBI, and 33 stolen bases.

[5] The Blue Jays invited Burns to spring training as a non-roster invitee in 2014,[7] but he did not make the team, returning to the New Hampshire Fisher Cats for the entire 2014 season.

[5] In the offseason, he appeared in 12 games for the Bravos de Margarita of the Venezuelan Professional Baseball League, and hit .205 with 5 RBI.

[20] On January 8, 2017, Burns signed a one-year, $650,000 contract with the Lotte Giants of the Korea Baseball Organization (KBO).

[22] He spent the 2019 season with the Triple–A Buffalo Bisons, posting a .275/.364/.470 slash line with 19 home runs and 63 RBI in 118 games.

[25] In his Dodgers debut, he recorded his first career MLB hit, an infield single off Texas Rangers starting pitcher Kolby Allard and also pitched an inning.

[30] The Dodgers added him to the post-season roster prior to Game 5 of the NLCS to replace an injured Justin Turner.