After separating Challah, is it OK to burn the piece of Challah while baking the batch of bread? If I did this, can I eat the bread that was baked with it? Does the oven need to cool off from the piece of Challah before baking the Challah?

One can burn the Challah if it is covered well with aluminum foil while baking the batch of bread. One should not bake the Challah uncovered in an oven.

The batch of bread may nonetheless be eaten provided it did not touch the piece of Challah that was baking while uncovered. It seems that one should Kasher the oven by lighting it at it’s highest temperature for 30 minutes. The rack in the oven where the piece of uncovered Challah was baking directly on top of it needs to be Kashered by way of Libun Gamur (blow torch Kashering till sparks fly or glows red hot).

See also:

What do we do with the dough removed for Hafrashas Challah?

Sources:

Rema to Yoreh Deah 322:5. Shulchan Aruch Admor Hazaken (Orach Chayim 447:9) – since the bread is כחוש – thin, it will not make the other דבר כחוש become forbidden in the oven בדיעבד.