Scene 67: Alone in Dumbledore's Office
1. Recognizable Room
This scene describes the physical features of the room (it's circular and has portraits of former headmasters) as well as Dumbledore's personal touches (curious little instruments). This lets Harry (and the reader) immediately know where he is when he enters Tom's memory later in the book. Harry recognizes the circular office (so he knows he's in the headmaster's office), but doesn't see the instruments (so he knows that Dumbledore isn't headmaster).
2. The Sorting Hat
A major topic in this book is exploring the similarities between the protagonist and the antagonist (ie. Harry and Voldemort). Voldemort was in Slytherin. Should Harry be in Slytherin? At this point in the book, the other characters are at their peak of suspecting and accusing Harry (see: the Hufflepuffs' library chat). The Sorting Hat's judgement escalates this peak to its true apex. The hat is the ultimate impartial judge, thus, if it says Slytherin, we as the reader have to seriously consider that (as does Harry. Remember, there has been buildup to this Sorting Hat moment. Harry has worried about the hat twice already p153, p197). This escalation is the perfect setup for Dumbledore to clear all suspicion in the next scene.
Side Note: My favorite part of this scene is the irony in Harry's actions here. The very act of taking the hat off the shelf is exactly the sort of thing a Slytherin would do. You're not supposed to touch things in the headmaster's office... unless you're an "by-any-means" Slytherin.
But wait, there's more: then the hat tells him what he doesn't want to hear. A typical Gryffindor reaction would be to remain brave and stoic no matter what the hat says. What Harry does is more violent, self-righteous, and typical Slytherin: he rips the hat off his head (while it's still talking!) and then tells it that its wrong (even though it's literally the ultimate authority on house placement).
3. Baby Fawkes
Harry is at a low point in this scene. Things couldn't possibly get any worse. But then they do, in the most ridiculous way possible. A bird bursts into flames! It's so ridiculous it's hard not to laugh. And the reader could use some comic relief here (they haven't really had any since Lockhart's ineptitude sent a snake flying p194).
But more importantly, the burning bird gives Harry the opportunity to show that he's a nice guy. Harry's immediate reaction to seeing a bird burst into flames is not to run away, but to try to save it. Nice guy. It's usually important to keep your protagonist likable, but especially here. The entire previous chapter has been building up suspicion around Harry as a potential suspect, to which Harry has reacted by acting kind of rude and mopey. This "nice guy"-moment both redeems the rudeness and dulls the suspicion (which sets up Dumbledore to completely remove it in the next scene).
I guess you could also say that Fawkes's death and rebirth here could symbolize some other rebirth going on in the story, but that'd be pretty contrived. I can't think of anything.
