Yes, I agree that Armand would never retain that letter and must have burned it because it is the only proof of his heritage, a heritage that he obviously finds loathsome and disgraceful.
We get clues throughout the story that make it clear that Armand is a cruel slave owner. The narrator says that "under [his rule] his negroes had forgotten how to be gay." Life had, apparently, become more difficult for them during Armand's tenure as Master than it had been under "the old master's easy-going and indulgent lifetime." We know Armand's feelings, then, about blacks. In order to be so cruel, Armand must see them as subhuman, like animals, intended to work and obey. How could he ever face knowing that he is one of the individuals he so despises? In burning the letter, he could allow himself, perhaps, the chance to forget.
Moreover, his disgust with blackness is made evident when he addresses Desiree about his suspicions about her. He picks her hand off his arm and "thrust [it] away from him," as though it were something gross. When he tells her to leave, "He thought Almighty God had dealt cruelly and unjustly with him; and felt, somehow, that he was paying Him back in kind when he stabbed thus into his wife's soul." If he felt God was cruel in making his wife of mixed race, then he would surely be that much more victimized to learn of his own heritage. He knows the neighbors came to gawk at his child and that even the slaves were talking. How could Armand retain his social standing or his own personal sense of superiority if he allowed himself to accept that he is what he hates? Burning the letter is the only way to pretend it is not real.