Institutions are demanding that artists who have supported Mr. Putin in the past issue clear condemnations of the Russian president and his invasion as a prerequisite for performing. Others are checking their rosters and poring over social media posts to ensure Russian performers have not made contentious statements about the war. The Polish National Opera has gone so far as to drop a production of Mussorgsky’s “Boris Godunov,” one of the greatest Russian operas, to express “solidarity with the people of Ukraine.”

The tensions pose a dilemma for cultural institutions and those who support them. Many have long tried to stay above the fray of current events, and have a deep belief in the role the arts can play in bridging divides. Now arts administrators, who have scant geopolitical expertise, find themselves in the midst of one of the most politically charged issues in recent decades, with little in the way of experience to draw on.

“We’re facing a totally new situation,” Andreas Homoki, the artistic director of the Zurich Opera, said. “Politics was never on our mind like this before.”