Broken Pedestal: General Iroh to Zuko; after realizing he wants Zuko to 'redeem' the Fire Nation as Fire Lord by assisting in his father and sister's demise and becoming a puppet Fire Lord in all but name, Zuko no longer considers Iroh as a mentor. Even more so when he discovers Iroh has been subtly hindering his efforts to capture the Avatar all along.
Death from Above: Eventually, seeing as both Zuko and Azula can generate lightning and are practicing flight by firebending.
Azula does this against the plantbender controlling the seaweed monster during the invasion of the Fire Nation, striking him dead with a bolt of lightning from an airship.
Despair Event Horizon: Aang and Co. seem to be nearing this after Zuko and Azula's attack on the Western Air Temple leaves Appa seriously injured and Haru dead.
Did Not Get the Girl: Mai with regards to Zuko; Zuko's new cold and ruthless persona causes him to lose all affection for Mai, who he eventually comes to regard as simply 'Azula's friend'.
Even Evil Has Loved Ones: Both Zuko and Azula are horrified by the possibility of their father's death when he insists on trying out lightning redirection immediately after Zuko shows him how to do it.
Fate Worse than Death: What Azula (and perhaps both Zuko and Ozai as well) intends for her uncle after he escaped during the eclipse.
"If they ever found him, uncle was going to be made to explain himself. Preferably at knifepoint. Over lava."
Incest Subtext: In later chapters a few interactions between the royal siblings are implicitly not of familial love.
Kick the Dog: While being 'interrogated' by Aang and company during the invasion of the Fire Nation, Azula tells Sokka that Suki is being 'used' by her fellow prisoners and that she might actually be starting to like it, and implies that she is pregnant...not that Sokka actually knows of the strictly enforced rules in Fire Nation prisons.
"...if they want to think them monsters, who was she to correct them?"
Moral Dissonance: Kizu and Shori, Ran and Shaw's two hatched children, bond with and eventually come to consider Zuko and Azula as their new parents after their real parents die by the siblings' hands.
My Country, Right or Wrong: Unlike in canon Zuko refuses to believe that anything the Fire Nation did is wrong, and as such considers the mere thought of helping the Avatar and overthrowing his father as treason.
Not So Different: Azula considers the 'new' Zuko as a different 'her', and despite now being her equal, she does not consider him a rival at all, quite the opposite actually, and begins to bond and even empathize with her brother like never before.
Lampshaded at one point by the Boiling Rock's warden: "Damned if he didn't match the princess perfectly though."
Azula also considers Toph to be this ability-wise.
"...the little Earthbending girl was a prodigy. An Earth equivalent of her almost, and with her blindness to force her into adapting and improvising to see using her bending. Possibly the most powerful Earthbender, and only twelve."
Pet the Dog: Azula informs the imprisoned Mechanist all the benefits of formally becoming a Fire Nation colony - autonomy, good relations with the Fire Nation homeland, complete support in case of dire need, the Mechanist's son getting pardoned for when he gets caught being with the Avatar, etc. - as well as the consequences of opposing the Fire Nation when Sozin's Comet finally arrives. The man quickly submits to the Princess and is implied to have accepted her offer for his people to become Fire Nation colonists.
Royals Who Actually Do Something: Zuko commands the defense of the Fire Nation Capital against the invasion during the eclipse, making the invaders pay a heavy toll in blood prior to the eclipse, and all but stopping their advance during the eclipse; when Azula launches a counterattack after the eclipse with the Fire Nation Air Fleet, she is IMPRESSED by Zuko's achievements.
During the attack on the Western Air Temple, he uses a fire beam to seriously injure Appa in a failed attempt to prevent their escape...and indirectly causes Haru's death.
Superpower Lottery: Zuko gains the ability to wield blue fire after he rejects his uncle's influence, and later on Azula teaches him how to successfully generate lightning and how to fully utilize the properties of blue fire. He later secretly teaches her and their father lightning redirection.
While Zuko still lags behind Azula when it comes to lightning generation, he outclasses her in redirection; to make up for this and to simultaneously take advantage of this difference in ability between them, they perfect a tactic that resembles a game of catch - wherein Azula generates lightning at Zuko, he redirects it back to her, she redirects it back to him, and so on, allowing them to catch any foes between them in a literal crossfire of lightning. They intend to reserve its use for when they settle the score with the Avatar, but Ozai is still pleased nonetheless by the very nature of the tactic.
Zuko later learns how to use fire beams, aka the Combustion Man's laser-like firebending.
The Bad Guy Wins: The story seems to be leading to this, though unlike in canon it doesn't seem like the Fire Nation is going to burn down the continent when Sozin's Comet arrives.
Thicker Than Water: Zuko refuses to betray his father despite the fact that he (Ozai) burned him.