Occupy oneself with unimportant matters and neglect important ones during a crisis. For example, The account was falling through, but he was more worried about missing his golf game—talk about fiddling while Rome burns! This expression alludes to the legend that the Emperor Nero played his fiddle while watching the conflagration of Rome. [Mid-1600s]

To do something trivial and irresponsible in the midst of an emergency; legend has it that while a fire destroyed the city of Rome, the emperor Nero played his violin, thus revealing his total lack of concern for his people and his empire.