Zuko's scar is something he prefers not to think about, let alone look at or touch. He's no longer sees his scar as a mark of dishonor. He knows now that he didn't deserve it and is actually kind of proud of himself for speaking up against something that he knew was wrong. But still, it made him uncomfortable to talk about. And sometimes, very rarely, it catches him off guard when he sees himself in the mirror, usually in the middle of the night when he least expects it.
When others mention or talk about it, it was even worse. Zuko knows that people can see it and are curious about it, but he wishes that they kept to themselves. He usually evaded questions about it and was able to brush off most people who were bold enough to even ask.
But touching his scar was completely different.
The first person who had tried was Uncle Iroh. He was trying to comfort Zuko and when he reached his hand out to touch his cheek, the boy recoiled so hard that Iroh was apologizing profusely for even attempting to touch it. It was uncomfortable and awkward and Zuko felt awful about it. His uncle was just trying to comfort him, but the memory of it was too fresh. Even though the scar had healed, he hadn't. Uncle didn't attempt to touch him again.
Next was Song, the kind Earth Kingdom girl who helped heal Uncle when they were traveling together. They were sitting on her porch and she was trying to empathize with him. When she reached out to touch it, he caught her hand. It was a little awkward but then she went on to show her own scar which left him shocked. It was probably the first time he had sympathized with someone in a long time.
Katara actually did touch it. In the catacombs under Ba Sing Se she had offered to heal it and he hesitantly agreed. If he could rid himself of this scar, then maybe, just maybe he could finally rid himself of the shame and dishonor it carried. She had only touched it lightly and he was so scared at first. Completely terrified at allowing someone in to help him finally heal. In the end nothing ended up coming of her offer and afterwards… well Zuko didn't like to think about afterwards.
When he and Mai were alone together, it was the first time his face had ever been caressed, and it felt nice. He didn't mind it when she touched his scar, he knew that she didn't see him and his scar, she just saw him. She had no issues pressing her forehead against it or caressing his face but when her hand lingered just a little too long on his scar, he would gently and discreetly move it somewhere else. It was just a preference, he'd reassured her when she asked, nothing to do with her.
His mother had cried when he told her how he got it. She was understandably upset at seeing her son's face marked and felt responsible for it. It took a lot of apologies, hugs, and tears for them to come to terms that it was neither of their faults. She had touched it, holding Zuko's face in her hands, but he could see the hurt behind her eyes. He hated that it brought his mother pain, so he quickly changed the subject, taking her to the turtle duck pond instead.
The one person who ever really touches his scar, and he means really touch it completely uninhibited, is his daughter. As a baby Izumi loved the feel of it under her fingers. She maps out the edge of it and every single groove. She doesn't only touch his scar of course, she grabs his nose, his lips, his cheeks, anything she can get her curious little hands on. But she always ends up absentmindedly tracing his scar. When he would feed her or read to her or rock her to sleep, she would reach out to gently touch it. It never bothers, upsets, or scares her. She loves his scar. He wasn't exactly thankful for it, but he was delighted that for once, it brought comfort instead of fear, or pain, or pity. He also knew that eventually, she would find out where his scar was from, and he wasn't looking forward to that conversation. But for right now, he was content to sit with his daughter while she explores his face and he kisses hers back.