At the beginning, when 15-year-old Michael starts vomiting, he enters an alley or courtyard and sits down. Immediately after Hanna also enters the same space, a woman with a gray coat, black head covering, and blue-and-white umbrella can be seen passing the same entrance. When Hanna turns to Michael and says, "Hey, hey you," the same woman passes the entrance from the same direction.
When Older Michael first gets into his car, it is completely covered with dew or raindrops. In the following scene, the car is totally dry.
Arriving at the church during the cycling holiday, Hanna gets off her bike on the right side of the bike and walks towards the church. When she parks the bike against the wall in the next shot she is standing to the bike's left.
In New York, Ilana Mather says, referring to her tea tin, "Not quite like this, it had Cyrillic lettering". But Hanna's tin she's holding does in fact have Cyrillic lettering - one can notice the Russian word "Chai" ("Tea") written on it. It is clearly visible when Michael puts the tin on the table.
When Michael visits New York in 1988, the cab he is in is followed by modern-day cars including a 2000s GMC SUV behind all the period vehicles.
When Michael reads Hannah the Odyssey, he reads from the Robert Fagles translation, which wasn't published until 1996.
When Michael is making the tapes, a notebook where he lists his recordings between 4 August 77 and 10 October 77 is visible. One of the entries is: Heinrich Böll, Women in a River Landscape. This novel was first published in 1985, so Michael couldn't have possibly read it in 1977. (Someone probably confused this title with another novel of Böll's, "Group Portrait with Lady", about the life of a woman born in 1922, which came out in 1970 and was very well-received and popular at the time.)
In the street scene just before Michael and Hanna go on their bicycle trip (1958), a Volkswagen Beetle drives by which has amber turn-signals in the taillights; these were not introduced until the 1962 model year.
Michael's black Mercedes sedan is accurate for the time period--the 1980s--but the numberplate starts with a nationality code, which was available in Germany no earlier than 1993, and even then only used for newly-created number plates. Further, it ends with the letter H, signifying that at least 30 years have passed since the car was first registered. (The 'H' gets granted to classic vehicles in mostly original condition, along with a tax break to support the upkeep).
Since Hanna can't read at all when she starts with The Lady with the Little Dog, she would not grasp so easily that the small and capital 't' are the same letter.