Your question asks for a subjective, opinion answer rather than one based in fact. So I'm going to speculate and give you my opinion of what Armand did.
Of course, I think he burned that letter. It was the only evidence that he, and not Desiree, was the one with mixed blood. He convinced her that she was the "flawed" one, so much so that she disappeared and was never seen again. He must protect his own reputation and standing in the community, so the truth must never come out. Since the letter is the only thing that tells the truth, then it makes sense that he would burn it.