The Coke can from which Joe drinks in the 1985 time setting (outside Justice with Lou) has the label "Coca-Cola Classic", rather than "Coke" or "New Coke" (introduced on 23 April that year). The design of the can is from 2002, not 1985.
When Hannah is trying to hail a cab for Prior outside the Mormon Visitors' Center, the taxis that drive past them are from the 1980s but the clearly visible cabs in the background are from the 1990s/2000s.
When Louis takes Joe to his Alphabet City (tenement) apartment, he opens his door which is in a long line of doors down the hallway. Once inside, he suddenly has two large windows, front and back, where there shouldn't be windows because there are more apartments on either side of his.
In the scene between Joe and Hanna at the Mormon visitors center while the 2 characters are arguing it's supposed to be raining outside, but if you look close you can see sunshine in the background and a woman walking in shorts.