When Naomi and Arthur enjoy their "first date" in the middle of Grand Central Terminal, darkness is seen outside the windows as if its the middle of the night, even though they were in bright daylight moments earlier outside on 42nd Street.
When the semi-naked Arthur is talking to Naomi from the street, the onlookers behind Arthur (who are genuine passers-by, not extras) change from shot to shot and back again. There are two distinct groups of them, revealing that two separate takes were intercut to create the final scene.
When Arthur is talking to Susan's father in the Johnson Tower, and they put on the safety glasses, Burt's ear hook goes behind the ear and then on top of the ear and then back behind the ear.
When Arthur is talking to Susan's father in the Johnson Tower, and they look through the telescope pointed to Arthur's apartment, the telescope has no eye piece attached.
In their first encounter, Arthur accidentally shoots Burt with a nail gun in his arm and shoulder. Then, when Burt is threatening Arthur with the saw, there is a bloodstain on Burt's shirt, but no hole.