During a time when the church and the state were equal in government, anyone failing to follow the state religion was burned at the stake.
Those who demanded the separation of church and state were considered heretics, and thousands who were caught discussing the issue were burned at the stake.
Because of this, whenever there was a secret debate on religious freedom, the subject was referred to as “the burning question.”
We often refer to an important issue as “the burning question” of the day.