General manager Rick Hahn said Cease has not tested positive for COVID-19.
"However, out of an abundance of caution, or arguably the appropriate amount of caution, we are going to keep Dylan away from the club until he can be cleared, based on the fact that he does have a couple of symptoms at this time," Hahn said.
"The length of Dylan's stay on the injured list is still to be determined. We're obviously hoping ... that it will be a brief one."
Right-hander Zack Burdi was recalled from the team's alternate site before its matchup with Cleveland. The 26-year-old Burdi made his big league debut last summer, going 0-1 with an 11.05 ERA in eight relief appearances.
Cease, 25, has no record and a 3.86 ERA in two starts this year. He permitted one run in 4 2/3 innings during Sunday's 10-inning loss to the Royals.
Cease's next turn in the rotation is scheduled for Friday at Boston. The team is hoping he can go, and coming up with a plan in case he isn't cleared to pitch against the Red Sox.