(at around 1h 3 mins) During the Quidditch Match, and the celebration afterwards in the Gryffindor Common room, you can see that the students are chanting "Weasley is our king!". However, the audio track we hear is "Weasley! Weasley! Weasley!" over and over.
(at around 36 mins) During Professor Slughorn's lesson, the students are told to make a Draught of Living Death, the recipe for which "can be found on page 10". During the lesson, it is quite clear that Hermione has her book open somewhere in the middle (circa page 200).
(at around 48 mins) When Harry opens the prince's book after the scene which Ginny takes the book from him and reads aloud "This book is property of the Half-Blood Prince" the camera goes through pages of the book and there's a page in which we see "Sectumpsempra. For Enemies". The first word is "Sectumsempra" not "SectumPsempra".
(at around 1h 55 mins) After Dumbledore shows Harry Tom Riddle's diary he tells Harry the diary is a Horcrux but he should have said 'was' as the diary was destroyed in Harry Potter and the Chamber of Secrets, meaning it is no longer a Horcrux.
(at around 57 mins) During Harry's first visit to the Weasley's house in Harry Potter and the Chamber of Secrets, a close up of the clock shows an option for the family to be at the dentist. At a Slug Club meeting, Hermione has to explain to the group what a dentist does, because that occupation doesn't exist in the wizarding world.
(at around 2h 5 mins) After Harry scoops up a sip of potion from the basin, there is still a scoopful left in the basin. However, after Dumbledore drinks it, the basin appears to be empty and Harry grabs the locket from the empty basin.
(at around 1h 45 mins) When Harry and Slughorn go visit Hagrid, Hagrid sits with a wine bottle and an earthenware goblet and Slughorn has only a goblet on his side of the table. At the end of the visit, Slughorn's goblet magically appears right next to Hagrid's goblet on his side of the table.
(at around 36 mins) When Harry's potions class is brewing the Draught of Living Death, right as Seamus' potion blows up in his face, Dean Thomas' head instantly changes position, clearly as a result of a cut.
(at around 8 mins) When Dumbledore puts the Muggle house "back in order", the clock clearly shows the time as 10:35 p.m. However, the waitress Harry meets at the sandwich shop said she'd be off at "11pm" but Harry and Dumbledore see her and her companion closing up the shop before they leave the train station.
The potions book which Harry gets from Slughorn's cupboard has the words "This book is property of the Half-Blood Prince" clearly handwritten on the first page. In a later shot the handwriting changes to a more print-like font.
(at around 1h 4 mins) When Hermione runs downstairs and she's sitting with the birds flying around her we can clearly count five birds. Later when Ron comes in with Lavender and she sends the Oppugno spell to Ron, very fast we can count six birds and six birds hit the door behind Ron.
(at around 5 mins) When Harry and Dumbledore stop and stand side by side before approaching the house where Horace Slughorn is hiding, the top of Harry's head is below Dumbledore's shoulder when they show them from behind. In the next scene we see them from the front and Harry is now clearly much taller as his nose is at Dumbledore's shoulder and they do not appear to have moved.
(at around 50 mins) In the Three Broomsticks pub, Slughorn spills some of his drink on the table and on Hermione. But a minute later when we see the table filmed from the side Slughorn was on, the table is dry, with no one taking out a wand to magically clean it up.
According to Harry Potter and the Order of the Phoenix, you need to know the purpose of the Room of Requirement to be able to access it. Harry and Ginny were looking for a place to hide the book, so the Room allowed them access.
In 1 or 2 scenes in the dining hall, Lavender Brown is shown sitting at a different table across from Ron Weasley. They were both in Gryffindor, so they should be at the same table. However, it is not uncommon for students to be seated at other Houses' tables. Lavender was avoiding Ron and chose not to sit with Gryffindor.
(at around 2 mins) Although the books take place in the mid-1990s, no such chronology is given in the movies, allowing the Millennium Bridge in London (built 1998-2000) to appear. On a similar note, the movies' biographical or age information about certain characters may vary from the books, but is not an internal goof.
Some have suggested Aragog is a great deal smaller in this movie than he was in Harry Potter and the Chamber of Secrets, but this only true to an extent, as spiders generally shrivel up pretty fast after dying. Add to this the fact that the persons around him this time around are quite a bit taller, plus the angles at which we saw him in his first appearance really complementing his statue while alive, and we have a very reasonable change which doesn't indicate any mistake or error by the filmmakers.
(at around 1h 55 mins) In theory, no one can Apparate into or out of Hogwarts as Dumbledore does in the last act. However, Dumbledore has secret information to override that particular magic. In fact, Harry points out this seeming problem and Dumbledore says, "Well, being me has its privileges."
When Harry first has the Half Blood Prince's potions book, his signature is in the front of the book. When Ginny takes it from him, she reads his signature from the back. However, the book may have been signed at the back and the front as later in the film we see the words written in two different styles suggesting multiple instances.
(at around 8 mins) When Dumbledore first introduces Harry to Professor Slughorn, Slughorn's robe changes color more than once. However, this is due to the fluorescent lighting outside of Slughorn's home.
Dumbledore has Riddle's diary in his possession. At the end of Harry Potter and the Chamber of Secrets, Dobby dropped the book to rescue Harry from Lucius Malfoy's wrath. The book was left on the floor, meaning that someone could return the book to the headmaster's office.
In previous films the Weasley home is surrounded by trees, while in this movie it is located in the middle of a huge empty field, so many have wanted to label this change as a goof (even if this clearly is a deliberate artistic change by the producers). However, we must bear in mind that the residents at the Burrow are a family of _wizards_! Hence, the work of removing some trees (or even transfiguring?) and boosting the growth of crop in the course of a winter shouldn't seem far-fetched at all in that context.
(at around 1h 40 mins) After Harry consumes the Felix Felicis, the vial seems to be still unused while Harry is talking to Ron and Hermione. However, when he is lowering the vial from his mouth, Harry can be seen replacing the stopper, giving the impression that the vial has not been used.
(at around 2 mins) During the destruction of the Millennium Bridge: to begin with the bridge is full of people, including a lot of school children, walking across. As the bridge begins to wobble, people start to trip up and fall down. When we see the bridge from above, as it is destroyed seconds later, there aren't any people on it.
(at around 1h 45 mins) In the scene with Hagrid and Professor Slughorn, you can see them both moving their hands when talking. Slughorn's hand often disappears/seems to go right through Hagrid's arm. They were filmed separately to exaggerate Hagrid's height.
(at around 5 mins) At the beginning when Harry and Dumbledore first disapparate, there is a full shot of them walking up to the entrance of the house. In this full shot, you can see Dumbledore's shoes sticking out of his robe as he walks, only he is actually wearing those blue bags on his shoes, the ones they wear in between takes so as not to damage the set floors.
(at around 49 mins) In the Weasley Brothers' shop scene, when they are showing Harry the Instant Darkness powder, an FX layer over the case holding the powder (meant to darken the case from a dull gray to true black) is clearly visible, extends beyond the bounds of the case and moves a bit during the shot.
(at around 36 mins) The "Draught of Living Death" page in Harry's book of "Advanced Potion Making" skips from step 8: "Chop three Valerian Roots in to small squared pieces. After cutting place it in a beaker with water. Leave it to settle for 7 minutes", to step 10: "Carefully pour the Sopophorus Beans' Juice into the cauldron making sure that ONLY the juice of 12 beans is used", without a step 9.
(at around 36 mins) The contents of Harry's book of "Advanced Potion Making" says "How to Prepare the Draught of Living Dead [sic]", whereas the actual page's heading is "Preparation of the Draught of Living Death".
(at around 55 mins) Several times throughout the movie, Harry's glasses clearly have no lens in them. They are just the frames. This is clearly evident when Harry and Ron Weasley are lying in bed and Harry is looking at the map showing the foot prints. The rest of the scenes show the glasses having lenses in them.
The goof items below may give away important plot points.
(at around 2h 10 mins) In the establishing shot right before Snape casts Avada Kedavra the tower's safety railing is clearly visible directly behind Dumbledore. But when the camera cuts to Harry's perspective of Dumbledore being hit by the spell, the railing has disappeared and Dumbledore falls right off the edge of the tower where the railing had been.
In the chase scene in the Corn field when the Death Eaters attack the Burrow, after Ginny and Harry make a stand before the Order arrive, Harry can clearly be seen pointing his wand. As the Order arrive and form the rest protective circle, he is seen pulling the wand out of his pocket.
(at around 1h 45 mins) When Hagrid is mourning over Aragog's dead body, Harry stands next to him. If you look closer, you'll notice that in faraway shots, Harry's head is clearly above Hagrid's elbow. However, in the closer shots, his head is below Hagrid's elbows.
(at around 2h 15 mins) At the end of the movie when everyone is holding their wands skyward in tribute, many have thought to see Hagrid holding someone's else's wand up. But that hand, if you look very closely, is not of Hagrid, but of a nearby student.
(at around 2h 15 mins) Near the end of the film, when Harry is lying on the ground outside Hagrid's cabin, Snape kicks Harry's wand away from his hand. When he does so, a square of neon tape is clearly seen, almost right in the middle of the shot, where Harry's wand was placed, so that actor Alan Rickman would know where to kick.