Chamberlain was hurt during Montell Owens' short run in the second quarter when he was caught up in a scrum near the tackle.
Trainers appeared to be treating Chamberlain's knee before carting him to the locker room.
Tanner was hurt blocking downfield after a short reception by Travaris Cadet. He could not put any weight on his left leg and trainers appeared to be looking at his ankle or foot before he was carted away.
Chamberlain was signed in free agency from St. Louis, where he played under new Saints defensive coordinator Steve Spagnuolo.
Tanner is in his third Saints training camp.