Burns, who has been playing at Triple-A Nashville, hit .234 with 12 RBIs in 73 games with Oakland this season. Eibner batted .231 with three homers and 10 RBIs in 26 games for the Royals.
Hill is 9-3 with a 2.25 ERA in 14 starts and hasn't pitched since July 17, when he threw only five pitches before leaving due to the blister.
Hill was scheduled to start Sunday but was scratched after experiencing discomfort during a throwing session Friday. Right-hander Sonny Gray will take Hill's turn against Cleveland.
Left-hander Dillon Overton was recalled from Nashville to start Saturday night against the Indians.