When Hermione first appears in the movie she has a newspaper. The picture of the Muggle family does not move even though it is in a Wizard newspaper. However, one must assume that because it is, in fact, a Muggle family that they would not have had a Wizard picture taken of themselves, and the picture is a Muggle picture, hence the subject's stationary positions. It can be seen that the headline picture, however, does move.
When Hermione stuns Harry to disguise him from the Snatchers, she takes his glasses off him and puts them in her own pocket. When Harry and Ron are put in the cellar at Malfoy Manor, Harry takes his glasses out of his pocket.
When Harry is preparing to jump into the lake after the sword of Gryffindor, he submerges, wearing his glasses. They are present as he drowns. After returning to the surface, all is blurry and he is no longer wearing them. Harry picks them up off the ice, from a position nowhere near the hole he dove through.
When Hermione and Harry are sitting on the rocks on the cliff the camera behind Harry shows Hermione putting down a book closed, when the cameras switch to behind Hermione it appears that the wind has blown the book open but the cameras switch back to show the book closed again, and switch one final time to see the book yet again open, flapping in the wind.
In the first scene at Malfoy Manor, Voldemort asks Pius Thicknesse for his opinion on the current situation. In the first shot of Pius, the ministry-pin on his tie is seen with the M vertical. In the shot of him replying, the bottom of the M is tilted slightly to the right seen from camera POV. In the shot of him reacting to Voldemort saying he will prove useful, the M is vertical again.
The piece of paper on which Xenophilius Lovegood draws the Hallows symbol changes between shots. When he is drawing the symbols there is a smudge on the paper, but when the shot pans away and he is talking to Harry, there is no smudge on the paper.
After Fleur transforms into a replica of Harry she (as Harry) is seen standing with her shirt off wearing a bra. The bra's cup size is far too small and positioned far too high on her chest to have been worn while Fleur was still herself.
In the third book, Prisoner of Azkaban, Harry visits Lupin in his office where he sees a Grindylow. This does not happen in the third film. This creates a continuity error as Harry could not possibly know the answer to Lupin's question (What creature sat in the corner the first time Harry Potter visited my office in Hogwarts?) after they return to the Burrow following the Battle of the Seven Potters.
When Hagrid and Harry emerge from the tunnel during the chase scene, they're on the North side of the Dartford crossing.
The Queen Elizabeth II Bridge is on the left, and traffic is seen going into the tunnel. This would mean the traffic is heading South.
This is unusual because the bridge usually carries the South-bound traffic, with the North-bound traffic split between the 2 tunnels.
It's not unusual for one of the tunnels to be closed at night, so it's possible the production took advantage of this and created traffic where they could put a camera crew in a position that would obstruct the normal flow.
When George and Fred are talking to Harry in the Dursley's house, the twins go from being in the back of the room to the front. You don't hear or see them walk up either. However, the previous movies have established that these characters are fond of "apparating" short distances.
In Harry's flashback of the night his parents were killed, Voldemort is shown in his resurrected form opposed to his pre-death state of being. Neither the book nor the movie tell us exactly what Voldemort looked like the night of the Potters' murders. Both, however, allude to his changing appearance becoming less human.
After the Fred and George take the "Harry" polyjuice potion, they both immediately shrink to Harry's size. However, their sleeves and trousers legs still fit them perfectly, when they should now be far too long.
When Harry walks across the frozen pond, both in his shoes and when he is barefoot, there are no footprints left on the ice. The pond is covered with a light snow, that Harry brushes away to see the Sword of Gryffindor beneath the frozen surface. The snow would show any tracks.
When Hermione is reading Rita Skeeter's book, she tells Harry that she has discovered that the boy in the photo at Bathilda Bagshot's house is Gellert Grindelwald, she hands Harry the book. If you look closely, you can see there is a oval shaped hole in the front cover where the moving picture of Dumbledore would be.