Why didn't Harry Potter use the Resurrection Stone?
Why did Dumbledore give the Resurrection Stone to Harry Potter in such a convoluted way?
Harry could've used it instantaneously to revive professor Dumbledore, even if only momentarily, but he eventually threw it away. What were his reasons?
Because Dumbledore did not want Harry to bring anyone back - if we remember the tale of the three Hallows, it was exactly bringing back the dead that got the original possessor of the Stone killed.
[T]he second brother journeyed to his own home, where he lived alone. Here he took out the stone that had the power to recall the dead, and turned it thrice in his hand. To his amazement and his delight, the figure of the girl he had once hoped to marry before her untimely death, appeared at once before him.
Yet she was sad and cold, separated from him as by a veil. Though she had returned to the mortal world, she did not truly belong there and suffered. Finally, the second brother, driven mad with hopeless longing, killed himself so as truly to join her.
Harry Potter and the Deathly Hallows, chapter 21: "The Tale of the Three Brothers"
In the book, Dumbledore explains that he didn't trust Harry the Stone because he didn't want Harry to repeat his own mistakes - namely chasing the Hallows:
‘Can you forgive me?’ he said. ‘Can you forgive me for not trusting you? For not telling you? Harry, I only feared that you would fail as I had failed. I only dreaded that you would make my mistakes. I crave your pardon, Harry. I have known, for some time now, that you are the better man.’
Ibid, chapter 35: "King's Cross"
Dumbledore explains pretty much everything about his own quest for Deathly Hallows back when he was younger, and he explains that it was important for Harry to take things slowly, and gradually become worthy to be the possessor of all three Hallows - the true master of death, one that does not run from it.
‘Why did you have to make it so difficult?’
Dumbledore’s smile was tremulous.
‘I am afraid I counted on Miss Granger to slow you up, Harry. I was afraid that your hot head might dominate your good heart. I was scared that, if presented outright with the facts about those tempting objects, you might seize the Hallows as I did, at the wrong time, for the wrong reasons. If you laid hands on them, I wanted you to possess them safely. You are the true master of death, because the true master does not seek to run away from Death. He accepts that he must die, and understands that there are far, far worse things in the living world than dying.’
I am also not convinced that the Stone can bring back the dead - temporarily or otherwise. If we take the Veil in the Ministry to be an indicator, then once a person is dead - they're dead. Thus I think that the Stone doesn't bring the dead back - it simply conjures a spectre from a person's memories (see also this answer on SFF, and linked ones). This is supported by the fact that only Harry could see his parents in the Forbidden Forest when he finally used the Stone.
‘We are part of you,’ said Sirius. ‘Invisible to anyone else.’ Ibid, chapter 34: "The Forest Again"
Harry wouldn't be able to resurrect Dumbledore in any meaningful sense, because he wouldn't be able to learn anything new from him - much in the same way as Dumbledore he encountered in "limbo" was not really Dumbledore.
It should be noted that Dumbledore himself believed that the Stone was incapable of providing anything but the illusion of a person; we can see that from his commentary on "The Tale of Three Brothers", found it The Tales of Beedle the Bard:
As I have already noted in the commentary for “Babbitty Rabbitty and Her Cackling Stump,” we remain incapable of raising the dead, and there is every reason to suppose that this will never happen. Vile substitutions have, of course, been attempted by Dark wizards, who have created Inferi, but these are ghastly puppets, not truly reawakened humans. What is more, Beedle’s story is quite explicit about the fact that the second brother’s lost love has not really returned from the dead. She has been sent by Death to lure the second brother into Death’s clutches, and is therefore cold, remote, tantalizingly both present and absent.
J. K. Rowling also provides the following footnote:
Necromancy is the Dark Art of raising the dead. It is a branch of magic that has never worked, as [The Tale of Three Brothers] makes clear.
I think it is clear that the authorial stance on ressurrection is that it doesn't work in any sense, meaning the dead can't communicate with the living in any way. Note that this doesn't contradict with the existence of ghosts, who are real persons, because they aren't fully dead in the first place.
It's also quite symbolic that Harry possessed the Invisibility Cloak, which originally belonged to the humblest, and the wisest of the three brothers.
And then Death asked the third and youngest brother what he would like. The youngest brother was the humblest and also the wisest of the brothers, and he did not trust Death. So he asked for something that would enable him to go forth from that place without being followed by Death. And Death, most unwillingly, handed over his own Cloak of Invisibility.
Ibid, chapter 21: "The Tale of the Three Brothers"
Humility is a very important trait of Harry's character, and it was in no small part a consequence of Dumbledore's decision to have Harry grow up with Dursleys. This is why the Snitch had "I open at the close" inscribed on it and opened when Harry said "I am about to die" - Harry had to give up his life willingly.
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Why Harry did not use the Resurrection Stone?Originally Answered: Why didn't Harry Potter use the Resurrection stone to bring his friends (who died in the Siege of Hogwarts) back? Because the dead were never meant to be brought back to life. They would come back to life, but they wouldn't be alive.
Did Harry Potter ever use the Resurrection Stone?Dumbledore left Harry the Resurrection Stone (disguised in a Snitch) in his will, and Harry uses it just before he goes to battle Voldemort. However, he doesn't use it to try to bring the dead back for an extended period or anything.
What was the point of Harry having the Resurrection Stone?As Cadmus was arrogant, he wanted to further humiliate Death, which led to him asking for the Resurrection Stone. He wished for an item that would enable him to bring back the dead, so Death took a stone from the riverbank and used his abilities to transform it into the Resurrection Stone.
Why did Voldemort not use the Resurrection Stone?In his entire time of possessing the ring, Riddle either never knew its powers or status as the Resurrection Stone, or simply did not have any use for it, as he feared the dead and did not wish to bring back anyone.
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More answers regarding why didn't Harry Potter use the Resurrection Stone?
While Gallifreyan is not technically incorrect, he failed to provide an actual good answer to this question.
Particularly he's right when he refers that the stone doesn't actually bring people from the dead, but instead just a specter. This specter however is very real, and it retains the personality of the person the user would like to bring back.
With that said, Harry did use the ressurection stone, in order to communicate with James, Lilly, Sirius and Lupin. After a motivational speech he let go of the specters and went to meet Voldemort.
The scene can even be found on YouTube!