I've seen this belief (that Harry kills Quirrell) associated specifically with the book universe several times, in various places, so allow me to vent about it. While Harry kills Quirrell in the films, he doesn't in the books.
These are the wounds Harry inflicts on Quirrell:
>And Quirrell, though pinning Harry to the ground with his knees, let go of his neck and stared, bewildered, at his own palms – Harry could see they looked burnt, raw, red and shiny.
Quirrell raised his hand to perform a deadly curse, but Harry, by instinct, reached up and grabbed Quirrell’s face –
‘AAAARGH!’
Quirrell rolled off him, his face blistering too, and then Harry knew: Quirrell couldn’t touch his bare skin, not without suffering terrible pain – his only chance was to keep hold of Quirrell, keep him in enough pain to stop him doing a curse.
They are burns, but Quirrell's pain indicates they aren't actually third-degree burns. What's described isn't enough to kill a Muggle, let alone a wizard.
>Harry jumped to his feet, caught Quirrell by the arm and hung on as tight as he could. Quirrell screamed and tried to throw Harry off – the pain in Harry’s head was building – he couldn’t see – he could only hear Quirrell’s terrible shrieks and Voldemort’s yells of ‘KILL HIM! KILL HIM!’ and other voices, maybe in Harry’s own head, crying, ‘Harry! Harry!’
He felt Quirrell’s arm wrenched from his grasp, knew all was lost, and fell into blackness, down… down… down…
When Harry falls unconscious, it's important to note that Quirrell is still alive (and screaming). Dumbledore later makes it very clear that he was still alive when he arrived:
>I arrived just in time to pull Quirrell off you
And he clarifies (more or less) what happened to Quirrell:
>He left Quirrell to die; he shows just as little mercy to his followers as his enemies.
We got the last two pieces of this puzzle in Goblet of Fire:
>I sometimes inhabited animals — snakes, of course, being my preference — but I was little better off inside them than as pure spirit, for their bodies were ill adapted to perform magic... and my possession of them shortened their lives; none of them lasted long... [...] The servant died when I left his body
Which, combined with what Firenze says in the first book:
>all you need is to stay alive long enough to drink something else
It makes it clear that what killed Quirrell was the possession itself and Voldemort's abrupt abandonment of his body. It had nothing to do with Harry, nor with him clinging to Quirrell, nor with Lily's protection. Quirrell was already dying (which is why he needed unicorn blood), as had all of Voldemort's other vessels. In the book universe, Tom Riddle, not Harry Potter, was the one who killed Quirrell.