Carbon isn't the only thing that burns. Burning is an exothermic reaction between a fuel and an oxidant -- when metal burns (for example, when overheated in the forge), the metal itself is the fuel.
Yes, it's a mixture of iron(II) oxide (FeO), iron(III) oxide (Fe₂O₃), and iron(II,III) oxide (Fe₃O₄, magnetite).
Scale starts to form at (very!) roughly the same temperature as the critical temperature of steel, which is why many bladesmiths wrap their blades in stainless steel heat treating foil to prevent oxidation. However, scale doesn't fall off the workpiece on its own until about 1750F, much higher than the temperatures in a heat-treating oven.