Continuity mistake: When Harry finds Hermione crying at the bottom of the stairs, with the conjured canaries flying overhead, there are numerous leaves/debris on the lowest steps and near her feet, but in the shots looking down from the top of the stairs the area is clear. (This has nothing to do with the few leaves that blow about.)
Revealing mistake: Harry and Dumbledore apparate onto Professor Slughorn's street, and Harry says "I just apparated, didn't I?" As Dumbledore is answering, ("Indeed. Quite successfully too, I might add.") the shot pans out. Look at Dumbledore's "burnt" hand. The green glove they used to make his hand look blackened is visible on his fingers. It is also visible when he turns and starts walking away from the camera. It is not due to the lighting, as it is not visible in the next shot as they both walk away. (00:05:15)
Continuity mistake: In the scene before the Quidditch match, while Ron, Hermione and Ron are talking and Ron believes that Harry put something in his drink - in some angles Ginny is sitting directly next to Harry with their arms only an inch or two apart, in other angles Ginny's arm is nowhere to be seen, when they are sitting close enough that she should be visible.
Other mistake: In the film Harry Potter and the Half Blood Prince, Professor Slughorn (during one of the 'Slugclub' meetings) asks Hermione what her parents do for a living, she replies, 'They're Dentists.' Slughorn and the other wizards stare blankly, not knowing what Dentists are. However, in the film 'Harry Potter and the Chamber of Secrets' in the burrow, we see the magical clock that shows the family's whereabouts, and we see that there is a section marked 'Dentist'.
Other mistake: When Harry and his friends visit the Weasley's house the second time - when the Death Eaters attack - the constellation Scorpio is visible in the sky. Scorpio is a summer constellation. It shouldn't be visible in the winter. Furthermore, it probably can't be seen as far north as England (although I don't live there, so I'm not sure).
Continuity mistake: When Dumbledore and Harry first arrive in front of Slughorn's temporary residence in Budleigh, Babberton, Harry's height barely even reaches Dumbledore's shoulders (likely the actor's stand-in), but in the next shot facing them Harry's height compared to Dumbledore is back to normal, level with the Headmaster's eyes. (00:05:35)
Continuity mistake: At the Three Broomsticks, when Slughorn and Harry speak, Slughorn holds a silver mug with a high foam head, but in the shot facing Harry, before Slughorn actually spills the drink the mug's foam head is gone (visible at the bottom of the screen), only to reappear when he spills it on Hermione ,whereupon the same previous angle is shown again without the foam.