The bottom of a firepot doesn't burn out because the sweet spot (where melting temperatures can occur) of the fire occurs a few inches above the bottom. Since the coal uses an updraft to achieve decent forging temperatures, a lot of the heat is moved away from the bottom. Possibly the tray also helps act as a heat sink?
It may eventually oxidize through, but this should take place much slower than with thinner metals, and shouldn't actually melt. I did lose a loose screen over the tuyere one time when it got tilted on end by my coal rake.
At least that's what I think. Someone with more expertise can and probably will explain it better. My firepot is homemade refractory funneling in over a brake disc, and it takes a couple of years to slag away the portland cement in it so that I have to patch or replace it. In my experience from making casting furnaces, this refractory recipe would start to slag at anything much above 2300 degrees, or brass casting temperatures.