Functional Order: Fi/Se/Ni/Te
Dominant Function: Fi
Zuko just can’t get away from his intense emotions, no matter how much he tries to repress or deny them in himself. His compassion and desire not to hurt innocent people is what caused the rift between him and his father in the first place, and the unfair comparison between him and his ENTJ sister. Zuko did not want to “waste lives needlessly” by sending them into battle as pawns, and stood up for this in the war counsel, which upset his father because he dared to question and challenge their Te-dom techniques.
Auxiliary Function: Se
Zuko is very physical and leaps into things quickly, sometimes without thinking them through. He enjoy going undercover in his pursuit of Aang, and even helps break into a castle to free him, in pursuit of freeing his uncle as well. He regularly strikes out at people using his fists and knees or feet, in addition to his abilities. Others see him as reckless.
Tertiary Function: Ni
He doubles down on his low Ni, making finding Aang his “one mission” in life, trusting his instincts, and going after him hard, without ever once considering that his father never wanted him to find Aang in the first place, that he cannot trust those around him, and that he could pursue a different path for himself, other than to follow in his father’s footsteps.
Inferior Function: Te
Zuko was then sent out on a mission to find and bring back the Avatar (his father thought it was hopeless, and banished him), and he routinely doubles-down too hard on his inferior Te in an attempt to be a “ruthless leader.” But the cracks keep showing. He can’t force himself to be unfair, cruel, or to make detached decisions; he saves his uncle even if it almost costs him his own life, he allows the Avatar to live, and he feels things too deeply to just move past them.
Note: This argument has been taken from Funky MBTI in Fiction, where it is applied to the life action version. I haven‘t watched it jet, but the argument fits perfectly in line with his animated version.