(at around 31 minutes) in the Gryfindor common room there is music playing in the background. The song is "Boys Will Be Boys" by The Ordinary Boys, which was not released until 2005 but this film is set in 1995 a full 10 years before the song's debut.
(at around 1h 45 mins) When Lucius has Harry and company at wandpoint in the Department of Mysteries, Bellatrix Lestrange walks out from behind him. In the audio she is laughing maniacally and talking to Harry, but when you look at her, her mouth is barely moving. She mouths "He knows how to play" a full second after it is heard, and her mouth moves at a much slower pace. She also has a deadpan expression on her face, not the maniacal face you would expect her to have while laughing like that.
Neville says that he never performed an Expelliarmus spell before, but in one of the previous Dark Arts trainings (within the sequence where Filch, Draco Malfoy & co. try to enter the Room of Requirement) he clearly performs one.
(at around 9 mins) Mad-Eye Moody tells Kingsley to take point as they escort Harry to headquarters, but throughout the flight Moody is in the lead. During the flight over the Thames River - just before the ferryboat - Kingsley can be seen to Harry's left. Moody should have told Kingsley to guard their flank.
(at around 21 mins) During Harry's trial, Fudge pronounces the name of the court as the "Wizenagamot"; the real name being the "Wizengamot". This could be intentional, to further underline Fudge's incompetence as Minister.
When the Weasley twins leave Hogwarts after the fireworks show, a girl in the cheering crowd wears Slytherin uniform. But Slytherin house is an ally of Umbridge, and they would never celebrate her being humiliated.
(at around 34 mins) When Harry is asleep during his first night at Hogwarts, he is wearing a striped, light blue T-shirt. But when he has a vision of Voldemort and sits up in bed, his shirt is now solid dark blue with buttons.
(at around 4 mins) When Harry is fighting off the Dementors, he produces a patronus and his wand is pointing at the Dementor that attacked him. When the scene cuts to Dudley on the ground, Harry's wand is at his side. When it cuts back to Harry, his wand is pointing at the dementor again.
(at around 38 mins) When Harry reports to Dolores Umbridge's office for discipline, she straightens a single crookedly placed pen on her desk. Later in the scene, the pens are no longer all in alignment.
(at around 36 mins) In Professor Umbridge's first class, she gives them all books, then proceeds to tell the class what to expect. During her speech, several of the students's books - particularly Ron's - switch positions from being examined by the students to being flat on the desk. The position of the books repeatedly switches positions as the camera angles switch.
(at around 7 mins) When Mr. & Mrs. Dursley put Dudley into the car to take him to hospital, the hanging basket near the door has mostly white flowers. Later that same evening when they prepare to take flight, they are now red, not white.
In the veil room, when the Death Eaters have all but Harry at wand-point, Bellatrix is shown first holding Hermione hostage. Immediately in the next shot she is being held by a male Death Eater, and Bellatrix is shown later to be holding Neville at Wand-Point.
(at around 1h 50 mins) In the veil room in the Department of Mysteries, when Harry tells the others to get behind him, we see Ginny stand directly behind Harry with Ron to her right. In the next shot when the Death Eaters arrive and swarm over them, Ron and Ginny have swapped positions in the formation.
(at around 38 mins) When Harry has to write lines for Umbridge, you clearly see him put his hand on the desk, then when the shot changes to a front view his hand is down by his side, and then when it changes back it is on the table again.
(at around 18 mins) When Harry and Mr. Weasley are trying to operate the ticket turnstile Harry pulls their tickets out of his jacket pocket once. Then when the camera angle changes Harry can be seen pulling out their tickets a second time.
(at around 35 mins) During Umbridge's first lesson she uses her wand to write out the OWL acronym on the chalkboard. In the initial shot the three words take up the majority of the board. However, in the shots following, the words are considerably smaller and further to the right.
(at around 1h 12 mins) When Harry is talking to Sirius in 12 Grimmauld Place (the Order of the Phoenix Headquarters) he is holding his Christmas present but in the next shot he is holding a glass of wine.
(at around 1h 11 mins) In the exchange of presents at the Phoenix Headquarters, Mrs. Weasley is handing over two presents to Fred and George. Firstly, the red parcel is in her right hand and the green parcel is in her left. In the next shot, they are in the other hands.
(at around 29 mins) When Harry and friends are riding in a cart from the Hogwarts Express station to Hogwarts, in a shot from behind, Luna Lovegood's head is facing to her left. In the following reverse angle, her head is facing to her right.
(at around 13 mins) When Harry meets Ron and Hermione at the headquarters, Hermoine's hair changes position. In the first shot Hermione's hair is behind her left shoulder, and in the next it is in front.
(at around 17 mins) When Harry and Mr. Weasley go to the London Underground, as Mr. Weasley approaches a turnstile, a man in a brown coat carrying a newspaper swipes his ticket and passes through the gates of an adjacent turnstile. Then the shot changes, and the man with the newspaper passes through the gate again.
At 1:14.48, where Sirius is explaining the Black family tree to Harry at 12 Grimmauld Place, the face of Bellatrix Lestrange is shown adjacent to Andromeda (whose face is scorched off the tree). Seconds later, the face of Bellatrix is gone, and only the name is visible.
(at around 1h 45 mins) When Harry and his friends fly on the thestrals to London, they are coming from Hogwarts, which according to the novels is somewhere in Scotland. So if they are flying from the North of the UK towards Westminster in London, where the Ministry of Magic is located, there is no reason why they would fly over the Canary Wharf district, which is 5 miles east of Westminster, especially if they are in a hurry to save Sirius Black.
(at around 18 mins) Present in the scene where Harry is being escorted by Arthur Weasley to his hearing at the Ministry of Magic, the two find themselves in a tube station which contains barriers that one may gain access through by use of an Oyster Card. J.K. Rowling has gone on record to state that Harry was born on July 31st, 1980. This would mean that the fifth installment in the Harry Potter series would take place in 1996, for Harry's fifth year in Hogwarts would see him being 16 years old. As this is the case, Oyster Cards would not exist at this time due to the fact that they were first invented in 2003. This mistake is under the pretense that the Harry Potter universes' time line was the same as the real worlds, with the fictitious addition of the magical world.
(at around 1h 21 mins) In Harry Potter and the Goblet of Fire, when Harry is looking in the Pensieve, Igor Karkaroff accuses Barty Crouch Jr. of torturing Frank and Alice Longbottom. In The Order of The Phoenix, after a DA meeting, Neville tells Harry it was Bellatrix Lestrange. Both statements are in fact true, as according to the books Frank and Alice Longbottom were tortured by a group of Death Eaters, which included both Barty Crouch Jr. and Bellatrix Lestrange.
(at around 37 mins) Umbridge punishes Harry for an outburst where he claims that Voldemort killed Cedric Diggory in the previous film, Harry Potter and the Goblet of Fire. Cedric Diggory was really killed by Peter Pettigrew (Wormtail), but only after a direct order from Lord Voldemort. Voldemort can be said to be analogous to Adolf Hitler (and others alike), who did not personally carry out his killings.
(at around 36 mins) In Ms. Umbridge's Defense Against the Dark Arts lesson the camera pans out over the classroom after she has said something and there is the sound of a pencil being dropped on the floor, when students at Hogwarts only use quills. However, this could simply be a dropped wand, as this is the sound we hear when this happens elsewhere in the movie.
(at around 18 mins) When Harry and Mr Weasley are descending into the Ministry, using the visitors' entrance, only the inside of the phone box in the street moves. When the scene cuts to the interior of the Ministry, they're in a complete phone box. However, if you look closely you can see that there is in fact a frame moving inside the original phone box, clearly it is a double. Otherwise, guests would have been in danger of falling off.
(at around 1h 30 mins) After Hermione rings the bell on the bicycle handle, they pan to Grawp and he is now holding the bicycle handle and rings the bell. We never see Hermione give it back, or Grawp take it back. However, if you look closely, they are different handles.
It is known from the previous movie that memories can be extracted and viewed through magic, yet when Harry needs to prove his claims on his Wizengamot hearing, nobody thinks of doing this to Harry's memories of that night, or Dudley's, who, although a Muggle, knows about the magical world.
(at around 43 mins) When Harry sees the thestrals dragging the carriages, the others cannot see them. Luna then claims that he can only see them because he has seen death (Cedric, as he could not see them in previous years). Although JK Rowling said that he wouldn't have recalled his mother's death clearly enough because of him being just a baby at the time. he did watch Quirell die before him in Harry Potter and the The Sorcerer's Stone. So he should have been able to see Thestrals since, at the earliest, his second year at Hogwarts.
(at around 1h 45 mins) Bellatrix Lestrange has been imprisoned for 14 years, well outside any contact with society, so how does she know (or recognise) Neville Longbottom the second she sees him? He was barely one year old 14 years ago.
In the first film, Harry is conscious at the point when Quirrell dies and is able to see it as it happens, unlike in the book. Because of this, he should've been able to see the Thestrals from his second year at school onward.
Lucius Malfoy says to harry in the ministry of magic that the prophecy could only be retrieved by whom it was about so why did Voldemort not go and fetch the prophecy himself "as the prophecy was not just about harry it was related to himself" rather than send his followers and death eaters he was a very powerful wizard and it would have been easier to do that than what was carried out .
(at around 54 mins) At the Hog's Head, Zacharias Smith says the line 'The point is, where's the proof?' While he's saying them, the camera is on Harry and you can see him mouthing 'where's the proof?'.
(at around 18 mins) When Harry and Mr. Weasley are entering the phone box which leads to the Ministry of Magic, at the end of the street several pedestrians are already crossing the road, while the traffic lights for the street are green in the beginning, and switch to red later.
The goof item below may give away important plot points.
(at around 1h 35 mins) When Harry has the vision of Voldemort torturing Sirius, he is wearing his school uniform. However, when he, Ron, and Hermione are in Umbridge's office to use her fire, presumably minutes after, they are all wearing Muggle clothes.
(at around 1h 55 mins) During the final Battle in the Department of Mysteries, in the shot swooping over the battle, where Bellatrix Lestrange is heard laughing, you see Moody in the bottom right of the shot, and he is clearly dueling with two Death Eaters, but in the next shot a single Death Eater runs out of the rocks, and attacks Moody, yet Moody is not engaged with any other Death Eaters.