Perhaps in the fanfiction, Harry gives up because he becomes disillusioned with the wizarding world. He may see all the corruption and prejudice within it and decides that Voldemort's way of ruling might not be much worse. This could lead to a story where Harry actually joins Voldemort's side, not as a slave but as an 'advisor' of sorts, trying to change Voldemort from the inside. However, this is a very dangerous and morally ambiguous path for Harry to take.
Voldemort and Harry's relationship would be very strange. Voldemort might not trust Harry at first, even though he has given up. He would constantly be on guard, thinking it's a trick. And Harry, on the other hand, would have to navigate living in the Death Eaters' world. The other Death Eaters would also be suspicious of Harry, some might even try to kill him to prove their loyalty to Voldemort.
If Harry gives Voldemort the stone in fanfiction, it could be a very dark turn of events. Voldemort would likely use the stone to bring back his physical form more fully or perhaps gain some sort of immortality. However, some fanfictions might explore the idea that this act actually unites the wizarding community against Voldemort in a way that they hadn't been before. They might see Harry's sacrifice as a call to arms and fight back even more fiercely.
For Harry, he might be seen as a traitor by some, but he could also be seen as a desperate hero trying to make a deal with Voldemort. Hermione would probably be horrified and try to figure out a way to reverse the situation. Ron might be angry at Harry at first but then realize there's more to it. Voldemort, on the other hand, would be overjoyed and his ego would inflate even more.
Perhaps Dumbledore is being forced in this fanfiction. There could be some dark magic or powerful dark wizard threatening Hogwarts or the wizarding world. Dumbledore, in a desperate attempt to save everyone else, gives Harry to Voldemort as a sacrifice. But Harry, being the hero he is, could find a way to turn the situation around and defeat Voldemort from within his lair.
I'm sorry, but such a concept is rather strange and not in line with the original 'Harry Potter' lore. However, you can try searching on some major fanfic websites like Archive of Our Own (AO3). Maybe there are some creative writers who have come up with unique interpretations, but I haven't come across such a specific one myself.
Hermione would be devastated. She would initially think Harry had lost his mind. She'd probably try to find a way to rescue him, using all her knowledge and research skills. Ron would be angry at first, feeling that Harry had betrayed them. But then he would also start to think there must be a deeper reason and would want to help. Dumbledore, if still alive in the fanfiction, would likely be more understanding, believing that Harry has a grand plan that he's not aware of yet.
One plot idea could be that Harry gives himself to Voldemort in an attempt to understand the Dark Lord's mind and find a weakness from within. Another could be that he sacrifices himself this way to protect his friends and Hogwarts, believing that there's a way to turn Voldemort's victory into a defeat from the inside. Or perhaps Harry has a vision that shows this as the only way to truly end the war and he decides to take the risk.
One possible reason could be that J.K. Rowling, the author of Harry Potter, has strict control over the franchise and might have made some regulations that made it difficult for him to continue. Another reason could be that he simply lost interest in writing fanfiction about it. Maybe he wanted to focus on creating his own original stories instead.
One possible reason could be that Ginny realizes Harry is too preoccupied with his heroic duties and she feels neglected. In fanfiction, this is a common trope where the female character desires more attention in a relationship.