Factual error: At one point, Newt Scamander says he found one of his beasts in Equatorial Guinea. That country only got that name after independence from Spain in 1968; back in 1926, when the movie is set, the place was known as Spanish Guinea.
Other mistake: When Newt gives Jacob a suitcase full of eggs, he bumps into him, apparently swapping suitcases. If you watch carefully, the suitcases don't actually switch.
Character mistake: In her speech outside the bank, Mary Lou Barebone mentions "the wireless" as one of the wonders of modern technology, which is the British term. A New Yorker in 1926 would more likely use the Americanism "radio."
Factual error: In the final scene in the bakery, there is a fire visible in the oven. In bakery ovens, there is no open fire in the oven, it is in a closed cabinet below. Bread and cake are not grilled. (02:03:55)
Continuity mistake: After eating their meal they're sitting at the table talking with empty plates in front of them. As they continue to talk all the plates and flatware disappear.
Continuity mistake: During dinner at the Goldsteins', the dishes are empty from one shot to another.
Other mistake: A shot of Jacob and Queenie walking was used twice. Queenie is showed on the left side of the screen and Jacob on the right, and Jacob looks behind, then turns to look to his left, then turns more in front of himself. This happens while Queenie and Jacob are trying to find Newt's case, then find what cell Newt and Tina are in when they are trying not to be killed by Grindlewald in disguise.
Character mistake: Toward the end of the movie, the behavior of Newt is totally out of character - not just for him, but for any sane person. Newt tries to save Credence, and he has just gotten him to stop attacking and actually listening - when the other wizards blast what Newt knows to be essentially a traumatized and abused child to kingdom come. But Newt isn't the least bit distressed or mad, he just shrugs it off in a "shit happens, life goes on" kind of manner, catches Grindelwald for them and helps them save the world, happy as a flea and not giving a damn about the recent death of Credence. Only a complete and utter sociopath would be totally unmoved by the death of a child they were trying to save. A sane, feeling person (like Newt) should be at the president's throat spells blazing, cursing her and the whole American Wizard-hood to the nether regions of hell and not giving a blue damn whether or not they are exposed and in trouble, and whether or not Grindelwald goes free, because in that moment, in his eyes they would be hardly better than him. It is even more out of character since Newt is until that moment depicted as extremely protective of his creatures.