When Newman burns the letter containing the juicy Bellegarde family secret, he's committing to move on with his life. Of course, Newman has to end the Claire saga of his life in the presence of Mrs. Tristram, his longtime pal and confidante.
She gets the final say:
"I suppose there is no harm in saying that you probably did not make them so very uncomfortable." (26.34)
Just to be clear, Shmoopers, that's a bit of a burn. The Bellegardes have left a gaping Claire-sized hole in Newman's heart, but he hasn't really affected them that much.
Newman's a good guy, though. He wouldn't really blackmail the Bellegardes, would he? But still, he can't help checking to see if the letter is totally burnt up after Mrs. Tristram drops her truth. Hey, it's human nature.