What is the source of this practice? Is it a Minhag or Halacha? What are the reasons it must be burnt?
It is a Halacha well documented in the written Torah, the Mishna and the Jerusalem-Talmud and then codified by all Poskim.
Torah: Bamidbar 15:20:
ראשית ערסתכם חלה תרימו תרומה כתרומת גרן כן תרימו אתה
As the first yield of your baking, you shall set aside a loaf as a gift; you shall set it aside as a gift like the gift from the threshing floor.
Mishna: Masechet Challa
Talmud Yerushalmi: Mas. Challa
Rambam: Ch. 5 of Hil. Bikurim etc.
Shulchan Aruch Yoreh De'ah section 322 - 330
That said, nowadays it may possibly be Rabbinic under certain circumstances.