The correct answer is C) explosions in the boiler furnace. The primary function of a flame safeguard system on an automatically fired auxiliary boiler is to prevent explosions in the boiler furnace. The flame safeguard system monitors the boiler's burner and will automatically shut off the fuel supply if the flame is extinguished or not properly established. This prevents the buildup of unburned fuel in the furnace, which could lead to an explosive condition. The other answer options are incorrect because they do not specifically address the primary purpose of the flame safeguard system. Option A) refers to preventing dry firing and overpressure, which are separate safety concerns. Option B) is too broad, as the flame safeguard system is specifically designed to prevent explosions, not just uncontrolled fires. Option D) is also not the primary function, as the flame safeguard system is focused on preventing furnace explosions, not overheating of pressure parts.