Why should something like a cast-iron pan get warped, discolored, or damaged due to the heat? It seems incorrect in the absence of an explanation with a proper mechanism.

Metal object can warp when heated non-uniformly. The hot parts expand more than the cold parts, which causes the warping. However, in an oven the object should be heated slowly and reasonably uniformly. If an object is made of two different metals, then it can warp due to the different expansions of the two metals, but that again doesn't apply to a cast-iron pan.

Discoloration makes even less sense. Cast-iron pans are black because they are covered with carbon. This is also what gives them their non-stick properties. That carbon is deposited initially by spraying the pan with vegetable oil and then heating it to a high temperature. That causes the volatile parts to vaporize, leaving only the carbon layer. This layer can withstand much higher temperatures than the iron.