The answering machine in Henry's apartment has "Digital Answering Machine" printed on it yet when the new message is played it sounds like a cassette tape rewinding before the message is actually played. Digital answering machines do not have tapes to rewind so there should be no taped rewinding sound.
During the chess game, the move "king's rook to e3" was announced. In fact the rook was moved to e1. The next move, "knight to c3, check," didn't put the king in check and Sam didn't move the knight there.
All of the books in the "bookstore" that Dr. Foster visits as he is trying to find Henry have library identification labels on their spines, revealing that the scene was actually shot in a public library (the 58th Street branch of the New York Public library in Manhattan, as it turns out).