Why doesn't Harry live with the Weasleys?

Question

As we know, whenever school ends, Harry is sent back to live with the Dursleys, where he suffers from abuse. He would much prefer to live with the Weasleys, and often visits there before returning to school. It stands to reason that they probably don't live that far from each other, since:

1. Both Harry and the Weasleys use Platform 9¾ to get to school

2. Ron was able to take Harry to the Burrow in a flying car within a reasonable amount of time

It's heavily hinted at that the Weasleys are poor. However, it's not like Harry would live there permanently. Three months isn't that bad, not to mention he's pretty well-off from his parents' money. There would be little to no extra expense on the Weasleys.

The only explanation is that Dumbledore goes out of his way to ensure Harry lives with the Dursleys. If so, why is this?

Answer 1

Living with his blood family gives Harry magical protection.

"While you can still call home the place where your mother's blood dwells, there you cannot be touched or harmed by Voldemort. He shed her blood, but it lives on in you and her sister. Her blood became your refuge. You need return there only once a year, but as long as you can still call it home, there he cannot hurt you. Your aunt knows this. I explained what I had done in the letter I left, with you, on her doorstep. She knows that allowing you houseroom may well have kept you alive for the past fifteen years"
_{-- Albus Dumbledore, Order of the Phoenix}

Answer 2

The answer to this question is a bit complicated. As the quote provided by rand al'thor states, Harry will be under a protection spell as long as he calls Privet Drive his home. The quote also states Harry could not be touched by Voldemort, although he is touched be Voldemort multiple times before he turns 17 (the protection wears off after he is 17). Maybe, the protection spell began to wear off because Harry didn't really consider Privet Drive his home anymore after arriving at Hogwarts (emphasis mine).

Perhaps it was because he was now so busy, what with Quidditch practice three evenings a week on top of all his homework, but Harry could hardly believe it when he realized that he’d already been at Hogwarts two months. The castle felt more like home than Privet Drive ever had.

If the protection spell was truly beginning to wear off after Harry's first year, there was no point in him returning to Privet Drive every year. Let's look at the evidence, and compare Harry touching Quirrell in SS/PS to Voldemort touching Harry in GoF (emphasis mine).

Harry felt Quirrell’s hand close on his wrist. At once, a needle-sharp pain seared across Harry’s scar; his head felt as though it was about to split in two; he yelled, struggling with all his might, and to his surprise, Quirrell let go of him. The pain in his head lessened — he looked around wildly to see where Quirrell had gone, and saw him hunched in pain, looking at his fingers — they were blistering before his eyes . . . “Master, I cannot hold him — my hands — my hands!”

Quirrell can't touch Harry without screaming in pain.

“You all know that on the night I lost my powers and my body, I tried to kill him. His mother died in the attempt to save him — and unwittingly provided him with a protection I admit I had not foreseen. . . . I could not touch the boy.” Voldemort raised one of his long white fingers and put it very close to Harry’s cheek. “His mother left upon him the traces of her sacrifice. . . . This is old magic, I should have remembered it, I was foolish to overlook it . . . but no matter. I can touch him now.” Harry felt the cold tip of the long white finger touch him, and thought his head would burst with the pain. Voldemort laughed softly in his ear, then took the finger away and continued addressing the Death Eaters.

Not only does Voldemort not scream, but he laughs. This could only mean the protection had worn off at some point between the first year and fourth year, and Harry was free to live with the Weasleys. He kept coming back to Privet Drive afterwards for plot reasons only.