When the inkwell is knocked over, the ink spills on the left page of the open file on the desk. In the next shot of the desk, both sides have ink nearly covering both pages and there is ink on other papers, too. Plus, the position of the blotter changes between the shots.
In one scene the Horla crushes a rose flat on a table, but in a subsequent scene, in a wider shot the rose on the table is visible and has not been flattened.
When Odette goes to lock her apartment door after her husband storms out, the camera moves in on the doorknob and key, then Simon pushes the door open. No corresponding plate or bolt for the lock and key is visible on the door edge, revealing there is no lock at all. This is a common shortcut used by set carpenters.
When Simon throws the box of art supplies at the Horla to prove he is standing in front of the mirror, you can see Simon's reflection in the sheet of glass used to create the invisible Horla.
When the painting of Odette is floating in the air toward Simon, wires can be seen above it that are manipulating it.
When Simon visits the prisoner Girot in his prison cell, after Girot attacks Simon and Simon strikes the prisoner who then falls down against the brick wall base and is leaning against it just before he falls to the side, you can see the wall bricks bend and squish which makes it obvious he bricks are foam.
At the start of the movie the date in Magistrate Simon Cordier's diary states 1886 as the year. 18-19 minutes into the movie, when he returns the picture of his dead wife and son back to the trunk upstairs, the camera pans to the right after it's open and there is a Teddy Bear.
Teddy Bears weren't around until shortly after the November 16, 1902 newspaper cartoon of "Teddy" Roosevelt's based on his bear hunting trip in Mississippi earlier in the month. ("Teddy" refused to kill a baby bear after its mother was killed). Steiff bears started at the Leipzig Toy Fair in March 1903.
When Simon writes the final entry into his diary, he shuts the book and begins placing it into the box along with the note. In reality, he would never just shut the book having just written in it with liquid ink. He would have first used the blotter, then closed the book. If not blotted, it would most likely smear.