L’chatchila, the challah portion should be burned in a fire by itself (רמ"א שכ"ב ס"ה ), as one may not benefit from it.
The poskim write that one should burn the challah in an oven before baking the bread (רמ"א שם, ש"ך סק"ח ). In those days, there was fire in the oven, so after the challah was burned, libun [application of extreme heat] was automatically performed, and the oven could be used. Nowadays, our ovens do not have fire inside, so if one were to burn the challah in an oven, the oven would need to be kashered.
In an oven. Some women wrap the challah in aluminum foil before placing it in the oven. This practice is best avoided since it can cause many pitfalls when the dough rises and pushes through the foil, as it often does, rendering the oven not kosher. If this happened, one should ask a rav how to kasher the oven.
Gas flame. Some burn the challah over a gas flame. This is also problematic, since the challah touches the grate, necessitating libun.
If burning it is not feasible, one can seal it in a plastic bag in a respectful manner and place it in the garbage (שו"ת מנחת יצחק ח"ד סי"ג ). This is the proper practice today, that we do not burn it in its own separate fire.
Some freeze the challah portion in a plastic bag and burn several weeks' worth together. One should make sure to mark the bag as hafrashas challah (שו"ת שבט הקהתי ח"ד סי' רע"ז ), so that it is not mistaken for gefilte fish or kneidlach [both real-life examples].
