Chicago also recalled right-hander Reynaldo Lopez from Triple-A Charlotte to start Friday's game against the Kansas City Royals.
The White Sox also outrighted left-hander David Holmberg to Charlotte.
Burr, 23, went 2-2 with a 1.89 ERA and 75 strikeouts in 39 minor league appearances between Single-A Kane County and Single-A Visalia this season. He was Arizona's fifth-round pick in the 2015 amateur draft.
Lopez, 23, is 6-7 with a 3.79 ERA and 131 strikeouts in 22 starts with Charlotte this season and will wear No. 40 with the White Sox. The 26-year-old Holmberg went 2-3 with a 3.55 ERA and 28 strikeouts in 33 games (six starts) with Chicago in 2017.
Chicago enters Friday's game last in the American League Central division and 13 games out of an AL wild-card spot.