Reading the Harry Potter books is a bit like sitting down and having someone tell you a really great bedtime story. There's something very personable and engaging about the novel, and this welcoming quality definitely comes across in its tone.
First off, we have humorous. The narrator is often pretty funny and tends to make little dry remarks and side comments to readers that make fun of the characters and their surroundings.
They had Potions that afternoon, which was an unqualified disaster. [...] Snape, standing watch with an air of vindictive pleasure, scribbled something that looked suspiciously like a zero onto his notes before moving away. (16.2.8)
"Er – Mr. Black – Sirius?" said Hermione.
Black jumped at being addressed like this and stared at Hermione as though he had never seen anything quite like her. (19.119-20)
Neither of these scenes are all that funny, of course. Harry failing and the entire Shrieking Shack sequence aren't hilarious as a whole. But the narrator here definitely makes us laugh – there's a kind of dry, amused delivery to the scene with Snape. You can picture Harry craning his neck trying to see what Snape, who's so over-the-top awful that it's funny, is writing. And Sirius's double-take reaction to the overly earnest Hermione is pretty darn amusing.
This is never done in a mean-spirited way, though. The humor here is just a part of the story and the world of the novel itself – the wizarding world is very fun and often very funny, so it makes sense that the overall tone of the novel reflects that. However, the world that Harry Potter inhabits isn't all fun and games, of course. There's a lot of darkness there. And while humor can help lighten up the darkness, the novel often lets the darkness stay as it is and speak for itself. Some things simply aren't funny.
And this brings us to another major part of the tone: sympathetic and empathetic. We're drawing a distinction between the two, so bear with us. First, we have sympathetic, which means that the novel invites us to sympathize with and feel compassion for many characters. How does the novel accomplish this? Well, it uses tone and imagery to create pathos, which is a fancy term that means that the writing create sympathy for someone or something. Here's an example in a scene with Hermione:
After ten minutes or so, during which the Firebolt was passed around and admired from every angle, the crowd dispersed enough and Harry and Ron had a clear view of Hermione, the only person who hadn't rushed over to them, bent over her work and carefully avoiding their eyes.[...]
Harry looked around at the cluttered table, at the long Arithmancy essay on which the ink was still glistening, at the even longer Muggle Studies essay [...] and at the rune translation Hermione was now pouring over. (12.5.51, 60)
In this passage, we have an image of Hermione being isolated from everyone else, and it sets up a contrast between the excited crowd of Gryffindors and the lonely, stressed out, and shunned Hermione. And the narrator adopts a sympathetic tone to Hermione here, through the voice of Harry, who clearly feels bad for her.
And now we come to empathy – empathy is sort of like sympathy on steroids, in a way. Empathy means that you totally identify with someone else's pain or suffering and that you understand it on a personal level. When you empathize with someone, you actually feel what they feel. We get some empathy in the novel's tone because of the fact that the narrator identifies so strongly with Harry. The novel is told primarily through Harry's point of view and we readers are invited to actually feel Harry's emotions, including his suffering, much more acutely than any other characters. The novel spends a lot of time analyzing Harry's thoughts and feelings and the tone reflects this interest in Harry's emotional state.