The Texas Rangers selected Burr in the 33rd round of the 2012 Major League Baseball draft.

[3] He made his professional debut with the Hillsboro Hops and ended the year with the Kane County Cougars,[4] posting a combined 4–1 record and 1.06 ERA in 34 innings pitched between both clubs.

On August 11, 2017, the Chicago White Sox acquired Burr from Arizona for international signing bonus pool money.

[6] Burr began the 2018 season with the Birmingham Barons and was promoted to the Triple-A Charlotte Knights in late July, appearing in 7 games out of the bullpen and posting a 1.08 ERA over 8+1⁄3 innings pitched.

[13] On August 30, 2022, Burr signed a minor league deal with the Tampa Bay Rays organization.

He spent the ensuing season with the Triple–A Durham Bulls, recording a 3.09 ERA with 23 strikeouts across 23+1⁄3 innings of work.