Zuko chooses to keep his scar
As we all know, Katara nearly heals Zuko's scar (in S2E20 "The Crossroad of Destiny") using the spirit oasis water from the North Pole, but then has to use it instead to heal Aang. (To be clear: if she'd used it on Zuko, Aang would've [passed]...)
After the series completes and the Fire Lord is defeated, Zuko becomes the new Fire Lord. Unlike his father, Zuko is known for having helped the Avatar in saving the world, and he is respected and loved. So, if the Aang Gaang so chose, they could pretty easily return to the North Pole and ask for more spirit oasis water to heal Zuko. Granted, the North Pole isn't going to be handing that out willy-nilly, but when the freaking Avatar and his friends who just saved the world showing up intending to heal the good Fire Lord... yeah, they would oblige.
So why does Zuko still have his scar in Legend of Korra? Simple: He chose to keep it. Why? For two reasons:
1. Although there is indeed more spirit oasis water, it is still a limited supply. Zuko's scar is not harming him in anyway, it is not cancerous. Healing it would be solely for the sake of vanity, and Zuko has grown enough to know there are better uses for that water.
2. After his entire journey, Zuko has used what's happened to him to grow. In his own words, "I've been through a lot in the past few years, and it's been hard. But I'm realizing that I had to go through all those things to learn the truth." (S3E12 "The Western Air Temple"). Zuko's scar is no longer a symbol of his lost honor, but it is now a symbol of his past journey which he had to go through to restore his own honor. Thus he is able to wear it as a badge of pride - "My father burned and banished me because I dared speak up for what was right. It helped me learn the truth of how the world sees us, and so I helped restore balance to the world."