The black flames did nothing to Naruto, the cloak he used acted as a barrier and then he removed it immediately. Naruto continued to use that arm to fight anyway.
I understand bro. Burns marks, but I think it was the wrong arm. It was his right arm that was attacked with Amaterasu. His left arm contains the "moon" seal given to him by the Sage of Six Paths which was wrapped in bandages on the movie preview.