Iron filings burn vigorously when sprinkled in the flame of a burner due to their high surface area and reactivity. The fine particles of iron provide a large surface area exposed to oxygen in the air. When exposed to the flame, the iron undergoes rapid oxidation, combining with oxygen to form iron oxide (rust) with the release of energy in the form of heat and light. The increased surface area of the filings accelerates the reaction, resulting in the intense and rapid combustion observed when sprinkled into the flame.