At the end of the opera scene (shot at the Academy of Music in Philadelphia) there is an exterior shot of the building. Reflected in the glass doors of the opera house is a neon parking garage sign. To be accurate, neon wasn't even discovered until 1898.
At the dinner in Paris, May says, "When we were in London, we could only manage one day at the National," but her lips actually mouth the words "one day at the Tate." The original dialogue would have been anachronistic, as the story is set in the 1870s and the Tate Gallery did not open until 1897.
Early in the story, Ellen and Beaufort enter Mrs. Mingott's home as Archer, May and Mrs. Welland are leaving. As May is heard saying "Good bye, Ellen," May looks in her direction as the words are said, but May's lips are clearly not moving
When the grandmother tells Newland and May that a marble likeness of her hands was sculpted by a famous artist, you hear her say the name of the artist, but her lips look as though they've said something completely different.
On the park bench in Boston, Newland Archer sits down with Ellen Olenska. As he sits, his gloves are off, and we see his wedding ring. The camera cuts away and then back, and his gloves are on. He then removes them.
When Newland is sitting outside Ellen's apartment waiting for his son, the butler closes the window. As he does so, the window catches the bright sun several times and is reflected on Newland's face. As he walks away, the sun is obviously (by shadows) coming from the opposite direction thus it would be impossible for it to be reflected by the window.
During the baptism of Newland's and May's child, the family priest blesses the child with "the Father, Son, and the Holy Spirit" instead of "the Father, Son and the Holy Ghost". The usage of this phrase only came about after the revision of the Episcopalian prayer book in the 1920s.