At the scene of a building collapse a cop tells the crowd it isn't a gas explosion because there is no smell of gas.
But natural gas is both odorless and colorless, adding odor to the gas was globally adopted after the New London School explosion of 1937.
So even if there had been a gas leak or explosion, the cop would not have been able to smell gas.
The torch of the Statue of Liberty is shown in its current design of gold covered copper. That version was installed in 1985. The original torch (currently on display in the museum) was a copper framework with glass and was illuminated from within, having been in that form since a major reworking in 1919.
When Newt is preparing to leave the ship at the beginning of the movie, an overhead shot shows a crowd of muddy footprints on the deck in front of him. All of these footprints seem to be from shoes with rubber treads, but such treads were not available until the late 1930s for mountain climbers and much later for everyday shoes. In the 1920s, all shoes would have had flat leather soles.
When Newt first goes to the sisters' apartment, a record of Ruth Etting singing "You're the Cream in My Coffee" is playing. However, this song was not published until 1928, and Etting didn't record it until 1929---though the film is set in 1926.
The light given off by street lights and automobile headlamps is a uniform bright white like that from a modern LED or halogen bulb. In 1926 these lights would have used incandescent bulbs whose light would be somewhat yellow in color and would vary in intensity from one fixture to another.
When Scamander is trying to capture the invisible beast in the department store, he is behind a red Santa Claus. Red Santa was not popular at the time. The movie was set 1926, Coca Cola didn't start to advertise red Santa until 1931. Although recent history has found that Santa was advertised wearing red and white between 1863-1886, Santa was mainly popular wearing a tan and green outfit. This means that the red Santa would not be in that department store.
In Mary Lou Barebone's initial speech at the protest rally, she refers to "the wireless" as one of the marvels of modern technology. That is the British term; a speaker in New York City in 1926 would have said "the radio."
When Newt chases one of his creatures through Central Park, the park is covered in snow and people are ice skating on a frozen pond. At any other time directly before or immediately after this scene and throughout the movie, there is no snow anywhere, and people are dressed for moderate weather.
When in Newt's case for the first time, Newt goes from wearing his jacket, vest and tie, to his vest and loose tie, to just his blouse and loose tie, back to his vest and tie again in just a few shots.
When Jacob is showing the bank manager his suitcase of cakes, the cakes are at first a bit tumbled about, unsurprising due to the case's rather rough treatment up to that point, and are rather piled up on the viewer's right hand side of the case. After a cut, the cakes' arrangement within the case change. No more piles.
While the main characters are walking under the elevated line, the train passing overhead is on the left track as if it would be in England rather than the track on the right side as it would be in the United States.
Manhattan didn't allow overhead wires, so streetcars used a conduit (an electrified third rail situated in a slot between the rails) to get electricity. Though the streetcars are correctly depicted lacking trolley poles, the conduit is not present.
Every man in the movie wears a hat when outdoors (common dress for 1926), except for the three chief male characters - Newt Scamander, Jacob Kowalski, and Percival Graves, who never ever do. While Newt and Graves are wizards, whose fashion sense can differ from that of Muggles, Jacob does not have this excuse. However, Jacob is quite socially awkward.
When Tina takes Newt to the Wand Registry Office at MACUSA, the nameplate on her desk says "Queenie Goldstein". However, if one looks closely in a shot of Newt standing in front of the desk while Tina searches for a form/quill, on the other side of the desk (facing towards Tina) is a nameplate that says "Porpentina Goldstein" (Tina's full name).
When the Immigration Officer in New York demands to see the contents of Newt's case, Newt activates a "Muggle Worthy" mechanism that hides the case's magical contents. When the officer opens the case, everything looks neatly arrayed as if Newt had just packed it--despite the fact that Newt had held the case vertically until he set it on the inspection table. An Immigration Officer would be accustomed to seeing the loose contents of a case that was held vertically having slid down to the bottom (now back) of the case--especially heavier items such as a watch or a magnifying glass.
The goof items below may give away important plot points.
Before going after the Obscurial, Newt leaves Tina the case with his creatures inside, as well as a book that details how to care for them. When Tina goes to follow him, she passes the case to her sister, but neglects to give her the book as well - it is visible on Tina's person in several scenes thereafter.
Grindelwald expects to be imprisoned rather than executed, and that he will escape. But we had seen earlier that the American wizards will execute major offenders. Considering the majority of his crimes were committed in Europe though, it would make sense that he would also be tried by the governments of those countries, who have a differing view of the death penalty than Americans do.
Though the wizards can freely apparate (i.e. teleport) at will, there are at least two instances where they could have--and should have--but don't: When Tina has to sneak Newt and Jacob into her apartment, and later when Queenie has to smuggle all three of them out of the MACUSA Building. This is not a plot hole as it was seen in the Harry Potter universe that it is possible to prevent apparation into certain places. Most prominently into Hogwarts, but also into the Ministry of Magic, which MACUSA resembles. Although this could explain why Tina did not apparate into her apartment, it seems unlikely they had such security in place, but it could be either to not alert her landlady in case she was a No-Maj by coming and going like that, or just that it is generally the normal procedure not to scare anyone inside the house (her sister in this case) by suddenly appearing by apparating. In the Harry Potter universe Ron's father apparates to the outside and walks in the door.