"Where do you think Harry would live after the war? I personally think he would stay at the Burrow for a while, but I can also see him coming back to Grimmauld Place. I know Harry sees the Weasleys as family, but I doubt he'd want to live there long-term. Harry's an introvert, and the Burrow isnât the best place for someone who needs time alone. But yeah, itâs my headcanon, so Iâm curious about your opinion on this."
I really like the idea of Harry returning to Grimmauld Place, though, I don't think it'll be immediately.
The first night after Voldemort is dead, he sleeps in his bed in Gryffindor Tower:
he turned away from the painted portraits, thinking now only of the fourposter bead lying waiting for him in Gryffindor Tower and wondering whether Kreacher might bring him a sandwich there
He might stay in Hogwarts for a bit to help out McGonagall and the others. While they're going through the bodies, preparing burials, until Mrs. Weasley insists it's bad for him, and he should come stay at the Burrow with them.
I actually think Harry would decline because I'm not sure how comfortable he'd feel there right after Fred died. How he could sit there, having died and came back, but Fred just died. He didn't come back. And, I don't think most of the Weasleys would really think it's Harry's fault, but, there will be some tension there. Like, I think Harry would feel really awkward about the whole thing and like when he's upset early in DH and in OotP, he'd want to be alone.
Harry tends to go quite and isolate himself when upset, so I see him finding himself back at Grimmauld with Kreacher, secluding himself from everyone for a bit and disassociating like early in OotP:
On the fourth night after Hedwigâs departure Harry was lying in one of his apathetic phases, staring at the ceiling, his exhausted mind quite blank, when his uncle entered his bedroom. Harry looked slowly around at him. Uncle Vernon was wearing his best suit and an expression of enormous smugness.
She [Hermione] threw herself into a chair, curled up, and started to cry. Harry felt dazed. He stooped, picked up the Horcrux, and placed it around his own neck. He dragged blankets off Ronâs bunk and threw them over Hermione. Then he climbed onto his own bed and stared up at the dark canvas roof, listening to the pounding of the rain.
Hary, when upset and depressed, tends to just, shut down. So I can see him going into this state in Grimmauld Place for a bit early after the war. (Except for a few visits to Andromeda and Teddy, since I think he'd take his godfather role very seriously).
That is until Ron and Hermione drop by to drag him to dinner at the Burrow.
During this first time at the Burrow is when Ginny probably corners him about their relationship, how that goes depends on what AU/ship you're writing. But even in canon (with the epilogue), I'm not sure how quickly they get back together. I mean, Ginny still goes back to Hogwarts and Harry would try to find his footing in his life, so, it's up to the writer.
Anyway, I think Harry is more likely to stay most of the time at Grimmauld Place but stop by the Burrow often enough in the year after the end of the war (after that first lunch/dinner Ron and Hermione drag him to).
This guy is going to use Sirius' bedroom again (like in DH) like the traumatized mess that he is.
Now, the question is, where is Teddy in this time?
We know during the battle he's with Andromeda, and I think Harry is going to take his godfather responsibility super seriously, as I mentioned. Harry doesn't trust adults in general, so I think, initially, he wouldn't really know what to think of Andromeda, so I think he'd be more inclined to visit her and Teddy before going to the Burrow â he feels responsible for Teddy and guilty over Fred. And Andromeda I think would feel familiar to him. I think she has the same arrogance as Sirius, she carrys a similar pain to Harry â she, too, lost everyone. They both only have Teddy.
(Yes, Harry has the Weasleys, and as much as he loves them and appreciates how Molly tries to make him feel included, I don't think he really feels like he belongs, yk?
The aftermath of Fred's death shows this well:
The Great Hall seemed to fly away, become smaller, shrink, as Harry reeled backward from the doorway. He could not draw breath. He could not bear to look at any of the other bodies, to see who else had died for him. He could not bear to join the Weasleys, could not look into their eyes, when if he had given himself up in the first place, Fred might never have died. . .
And after the war, he skips Ginny and Molly to get to Ron and Hermione:
He spotted Ginny two tables away; she was sitting with her head on her motherâs shoulder: There would be time to talk later, hours and days and maybe years in which to talk.
[...]
Everywhere he looked he saw families reunited, and finally, he saw the two whose company he craved most.
Ron and Hermione are the closest to family for Harry, not all the Weasleys. When he wants to reunite with family, it's Ron and Hermione, not Ginny and the rest of the Weasleys. (And Luna earlier, but that's because she is comforting to him in general, but he doesn't see her the way he does Ron and Hermione))
Eventually, though, he'd probably move out of Grimmauld and to a different place of his own. With who and when is more up to personal headcanon and whether you consider the epilogue canon or not.