I've been spending far too much time looking at ff.net HP stories.
So. In the course of this, I discovered a little subsubgenre where Snape or Dumbledore "rescue" Harry from the Dursleys before he turns eleven. Okay, I've found, thus far, three stories. Two have Snape not just rescuing him, but also raising him, once in concert with Narcissa Malfoy (Lucius appears to be dead), and the other only had one chapter, but appears to be Dumbledore raising him.
So. What would happen...
These stories either have Snape barging in and wresting the poor abused baby Harry from the Dursleys and the Cupboard of Horrors (okay, not something that would be *bad*) or have the Dursleys killed by Death Eaters two full years before Harry gets his Letter.
Let's be a little simpler. The Dursleys are hit by a lorry after dropping Piers at home after Dudley's eighth birthday outing. Both parents die immediately; Dudley is barely hurt at all. Harry is at Arabella Figg's. Dumbledore, who knows All (including how Harry has been treated - note that the first letter is addressed to "The Cupboard under the Stairs" - immediately floos to Arabella's house. With Petunia dead, the blood magic is over, and Harry needs some other form of protection and fast. He takes the child, still numb with shock and probably guilt (he knows that if he wants something really bad enough, it happens *and* you know he's wished the Dursleys would die a time or two by this point) and having lost the only family and home he's ever really known, back to Hogwarts, which can't help with the shock.
He probably puts a plate of sandwiches and a goblet of pumpkin juice in front of Harry, but I would doubt he could eat.
Remember, Dumbledore doesn't have experience dealing with kids under eleven.
He does know that, though, so he calls on the experts. Molly and Arthur Weasley. And Molly, once hearing the facts (raised as a Muggle by people who disliked him and treated him poorly, but those people now dead) would floo right over.
And I get this picture. Molly approaching little Harry, wanting to offer comfort, but not sure how he'll take it from a stranger. And Harry, Gryffindor that he's meant to be, doesn't react to her overtures with fear. Molly could have handled fear, understood fear. No. He reacts with *confusion*, not knowing what she's on about. Because you know the last time he was cuddled was the day his parents died.
Anyway, knowing the task she's set for herself, she takes him home with her. Probably Dumbledore arranges for her to get a small allowance for his upkeep from the Ministry or Harry's own vault - which she will try to refuse. And she treats him right away like a member of her family. Which means lots of food and lots of attention, but also chores and discipline. No spoiling, no being a young prince. And probably, lots of well-meaning attempts to get him to cry or mourn or something, which he wouldn't do.
And it would still take a lot of effort for him to trust any adults. And there would need to be extra protections around The Burrow.
And I have no idea how this would affect him at age eleven. Except that he would be aware of his own background at that point, and really not like Draco.
On a different note:
During GoF, Ron goes on and on about how poor he is and how all he owns is rubbish.
Why hasn't he noticed that his best friend is in the same straits? Yes, Harry's school things are new and fit him, but other than those, and the Weasley jumpers, everything is old and far too big, and his glasses are in poor shape.