The carbon rich area of the forge ( reducing atmosphere) will eat up any extra oxygen, and help with scale and decarb, but will add no carbon to the blade.
In the scenario given by Bo T, there would be no effect on the blade at all. It would be softer if anything, not harder.
Case hardening is done by packing the blade in carbon rich material ( bone and charcoal plus some stuff to supply a bit of oxygen) and sealing the container. The box is then heated for hours at almost welding temps. The carbon does not move into the steel, as it is a solid, and solids don't do that sort of stuff. The oxygen in the closed system combines with the carbon as CO and this permeates the skin of the steel....very shallowly. The carbon moves a bit deeper as it joins the steel, and the oxygen goes out looking for some more carbon in the charcoal to do it all again...over and over. It takes a good bit of time just to put a skin of hard steel on the soft metal.
Of course, if you grind or sand the steel , the hard skin of "case" is removed.