Harry Potter's hands burned Quirinus Quirrell because of the protective magic his mother left in him when she sacrificed herself to save him. This magic acted as a shield and caused Quirrell, who was possessed by Voldemort, to burn and blister upon physical contact with Harry.
He asks Dumbledore again about the stone and Dumbledore tells him that he arrived just in time to save Harry from Quirrell. Dumbledore adds that he spoke with Nicolas Flamel and they decided to destroy the stone. He explains also that Quirrell could not touch Harry because Harry was protected by his mother's love.
Harry then asks why Quirrell couldn't touch him; Dumbledore explains that the kind of love and sacrifice that Lily made in dying for Harry gives a person some protection forever. Quirrell, who is “full of hatred, greed, and ambition,” could not understand this love and “could not touch [Harry] for this reason.”
The reason why Quirrel couldn't touch Harry at the end of the first book is because Harry's mother Lily left behind a powerful magical enchantment when she died that protects him from Voldemort, who was possessing Quirrel at the time.
Why Didn't Quirrell Lose His Arm When He Shook Harry's Hand - Harry Potter Explained
Did Dumbledore know Quirrell was Voldemort?
As Headmaster Albus Dumbledore was a highly intelligent and powerful wizard, he was aware of Quirrell's true nature even though he maintained a relatively civil relationship with him. This was further evidenced by the fact he asked Snape to keep an eye on Quirrell.
Professor Quirrell was a temporary type of Horcrux while Voldemort inhabited his body, which is confirmed in the Wizarding World. It is possible that Voldemort created more temporary Horcruxes, such as the animals he possessed while in ghostly form.
Voldemort ordered Quirrell to kill Harry Potter, though Quirrell's attempts to do so were thwarted by Severus Snape and Hermione Granger. Harry penetrated the Sorcerer's Stone Chambers in pursuit of Snape, or so he believed, only to find it was Quirrell in front of the Mirror of Erised.
He remained alive because of his Horcruxes, but existed in a spectral form as his soul had been driven from his body. In this form, he went into hiding in Albania.
He then decided that the best thing that could ever happen in his life would be finding what remained of Lord Voldemort after the Dark Lord's defeat. Quirrell also believed he could learn even more from Voldemort, and no one would ever laugh at him again.
"... have mercy..." Lily Potter's death practically set in motion the wheels for the entire series. She died at her home in Godric's Hollow trying to protect infant Harry, whom Voldemort was hell-bent on murdering. Lily's last words were an ardent request to the Dark Lord to let her little boy live.
Dumbledore knew that Harry would survive and didn't set him up for death, as earlier clues in the series hinted at his understanding of Harry's destiny. Harry's survival against Voldemort's Killing Curse revealed that Dumbledore knew all along that the Boy Who Lived was never in true danger.
James is destined to die, as Voldemort never gives him an option to save himself. On the other hand, Lily has the choice to step aside — most likely because Snape had requested Voldemort to spare her life, in exchange for details about the prophecy.
While in school together, Sirius once tricked Snape into almost entering the Shrieking Shack while Lupin was there, transformed into a werewolf. James realised the danger and stopped Snape, saving his life; this is the incident Dumbledore referred to at the end of the first book.
The prevailing theory has been that Voldemort gradually lost his nose (and humanity) throughout his life due to his exclusive use of Dark Magic. The more Dark Magic he tapped into, the less human he became both figuratively and literally. This was only amplified as he split his soul to create the Horcruxes.
After attempting to murder Harry as a baby, Voldemort was left without a physical form and was severely weakened. Voldemort was forced to flee to Albania to avoid anyone trying to hunt him down and had to possess animals to keep a physical form.
Notably, Voldemort appeared to be entirely unaware of this split, as he later created a seventh Horcrux in his quest to have six, and continued to target Harry Potter despite him now holding part of Voldemort's own soul.
Pettigrew mixed up a potion for Voldemort using unicorn blood and the venom of Nagini. With this potion, they were able to create a new body for Lord Voldemort to finally return to. However, this body was not what Voldemort needed to return to his fearful leadership form. Instead, it was that of a scaly, hairless baby.
Why didn't Harry see the thestrals after Quirrell?
Because thestrals can only be seen by people who have seen death, and understood it. Harry was unconscious when Quirrel died, so he didn't see anything. The next time Harry witnesses someone's death is in Book 4, with Cedric Diggory's death in the Little Hangleton cemetery.
It was only after Quirrell failed to procure the Philosopher's Stone from Gringotts that he latched onto his body (making him a Horcrux) so he could “keep a closer watch”. With Voldemort's face now sticking out of the back of his head, Quirrell donned a purple turban to hide him.
Quirrell did not try to use Avada Kedavra right away because with the prior incantatio you can know what spells a wand made, and he still had to stay undercover, and using such spells condemns you to Azkaban.
Snape's explanation to the Death Eaters was that he knew nothing of Voldemort's involvement, but "saw only greedy and unworthy Quirrell attempting to steal the stone." It was confirmed in Deathly Hallows that Snape had been "keeping an eye" on Quirrell on Dumbledore's orders.
Under the influence of her father, Dolores grew up despising her mother and brother for their lack of magical abilities. She was sorted into Slytherin House at Hogwarts School of Witchcraft and Wizardry and hated her time at the school due to never being given any positions of power.
Seeing as he died at the end of the school year Voldemort left Quirrell early to mid-1992. If you go from mid 1990 to mid 1992 it aproxamently comes to two years. So Quirrell was possessed by Lord Voldemort for around two years.
