Harry's scar was not just a remnant of the curse that failed to kill him. Remember when Voldemort casted Avada Kedavra on baby Harry, it never worked. Harry wasn't affected by it in any way. It rebounded because of Lily's protection, killing Voldemort. A piece of Voldemort's soul, unwilling to die, bonded itself to the infant Harry. That's what left the scar.
From Chamber of Secrets, "Dobby's Reward"
‘You can speak Parseltongue, Harry,’ said Dumbledore calmly, ‘because
Lord Voldemort – who is the last remaining descendant of Salazar
Slytherin – can speak Parseltongue. Unless I’m much mistaken, he
transferred some of his own powers to you the night he gave you that
In Order of the Phoenix, "The Lost Prophecy"
‘I guessed, fifteen years ago,’ said Dumbledore, ‘when I saw the scar
on your forehead, what it might mean. I guessed that it might be the
sign of a connection forged between you and Voldemort.’
In Deathly Hallows, "King's Cross Station"
‘But ...’ Harry raised his hand instinctively towards the lightning
scar. It did not seem to be there. ‘But I should have died – I didn’t
defend myself! I meant to let him kill me!’
I believe his scar was absent in King's Cross Station because the Harry in King's Cross Station represented Harry's soul. Voldemort was finally separated from Harry's soul at that point. As Xantec pointed out however, his bodily scar remained. However, as we are told in the Epilogue:
The scar had not pained Harry for nineteen years. All was well.
When viewing the full history of his scar, it becomes obvious that it was more than just the result of a curse. Avada Kedavra never scarred Harry. To the casual observer, it would appear to be the cause. Dumbledore knew better. He knew that that scar was caused by the "unintentional Horcrux" Voldemort left behind. Dumbledore also would have known that making this common knowledge would be very dangerous to the effort to undo Voldemort's remaining soul, so he covered it up deliberately, saying the scar was an exception to Avada Kedavra's undetectable powers.
ETA by Slytherincess: A little blurb regarding Harry's scar being the point of entry of Voldemort's soul, and the fact that when Harry's scar hurts it's because that remnant of Voldemort's soul is trying to rejoin its master by leaving Harry by the means of original entry.
Q: Why does Harry feel pain in his scar?
J.K. Rowling: Well, of course the pain he feels [in his scar] whenever Voldemort's particularly active is this piece of soul seeking to rejoin the master soul. When his scar is hurting him so much, that's not scar tissue hurting him. That's this piece of soul really wanting to get back out the way it entered. It really wants to- it entered this boy's body through a wound, and it wants to rejoin the master. So when Voldemort's near him, when he's particularly active, this connection, (JN: Oh, my gosh!) it was always there. That's what I always imagined this pain was. Yes, so there you go.
J.K. Rowling - THE LEAKY CAULDRON - POTTERCAST'S INTERVIEW WITH J.K. ROWLING- 12.23.2007