Hi Wang. Sorry for your situation. I've been through something similar, and feel for you.
Legally, you no longer own the ring and have no say over its future. Once you gave it to your fiancee', it became hers, to do with as she wishes. If she doesn't designate a specific heir to receive it in her will, it will be designated to the legal next-of-kin (determined by your state's order of designation) upon her passing.
If she has a daughter of legal age, for example, the ring would pass to the daughter. If your fiancee' decided to gift it to another or re-gift it to you before her passing, she might possibly do that as well. I'm not as clear on that aspect, as the ring is part of her estate, and you've got heirs involved. RockDoc knows this stuff better, he'll probably chime in here after a while.
If your fiancee' is like my wife was (practical), she wouldn't want you to burn it. That benefits no one, and destroys a beautiful family heirloom.
You guys should talk about it. You'll feel better afterwards.