- While talking with Slughorn to persuade him to come out of retirement, there's a moment where Harry and Slughorn notice the ring Dumbledore is wearing on his finger. Slughorn is specifically mentioned to be frowning. He recognizes the ring — it was worn by Tom Riddle. Including in the memory of him asking Slughorn about Horcruxes.
- The first Potions lesson, Slughorn points out several cauldrons full of various potions he has prepared. Each of them ends up playing a role in this book. The Felix Felicis gives Harry the right circumstances to get a memory and keeps the gang safe from most spells during the battle; the Amorentia is a love potion, which Ron ends up accidentally ingesting, and likely the potion Merope used to get Tom Riddle to fall in love with her; and the Polyjuice Potion is how Malfoy got his hands on the supply to use on Crabbe and Goyle.
- Harry notices that the barman from the Hog's Head was putting something away, after finishing up a transaction with Mudungus Fletcher. That was the other half of the two-way mirror Sirius had given Harry the year before, and what enabled Aberforth to keep an eye on Harry over the course of Book 7. Also explains the familiar eye of Dumbledore Harry saw in his own mirror's shard.
- Speaking of said barman, when Dumbledore shows Harry his memory of the time Voldemort came to apply to work at Hogwarts and learns Dumbledore knows his underlings' identities, he tells Dumbledore that he is "omniscient as ever" and Dumbledore replies that he is "merely friends with the local barman." Assuming we're talking about the same person, they were more than just friends, as we learn in the next book.
- Snape knows exactly where Harry learned the Sectumsempra spell from. Not only because of his Legilimency, but also because Snape invented the spell himself. In Book 7, Lupin even says that Sectumsempra was kind of a signature spell for Snape.
- The revelation that Madame Rosmerta was Imperiused for most of the book had hints here and there, eventually mentioned by Dumbledore later. When Hermione thinks Rosmerta is in the back and getting more Firewhiskey, she's actually in the bathroom and waiting to hand the package with the cursed necklace to a Hogwarts student. And Slughorn mentions the mead he was meant to give Dumbledore for Christmas was from Rosmerta.
- "Severus... I need Severus." Dumbledore realized his death was near, whether by a Death Eater's hand or the curse he's suffering from, and was trying to make sure his plan could still be pulled off.
Foreshadowing / Harry Potter and the Half-Blood Prince