Of course we “turn the heat on,” in this modern age, when we adjust the thermostat to a higher thermometer reading, and thus start up the oil-burning or gas-fired furnace or the electric heater.
But in current slang, the “heat” is of a different nature.
At first, it would seem, it was that of the electric chair, called in criminal jargon the “hot seat.”
But then, approximately the time of the First World War, it became a slang substitute for the accepted colloquialism of three centuries standing, “to make it hot for,” that is, to make one extremely uncomfortable, as by the grilling of a district attorney.
More recently, the “heat” has been extended to include search for a person suspected of crime, and it may be even a master criminal who “turns on the heat” when grilling a subordinate.