Iroh doesn't dismiss Azula because she was "evil"
S02E01, the Avatar State, Iroh to Azula: [Clearly upset.] To what do we owe this honor?
S02E11, The Desert, Iroh to the Rough Rhinos: Colonel Mongke! What a pleasant surprise!
For the longest time, I thought this was the case, or at least how he preseives it. No, it wasn't that simple. Iroh doesn't dislike Azula because she was evil nor because she reminds him of Ozai, at least we can make sure of that now since he admits that the one who reminds him of Ozai was Zuko actually in his official book (i.e Legacy of the Fire Nation) -- and it's a fair comparison (even though, Azula might reminds him of older Ozai; what Ozai become is who Azula is). And most likely, the feelings he harbors against her aren't that of a righteous rage.
Iroh worked with the Rough Rhinos, who are mass murderers mercenaries, and was cool with them before and after his 'reformation', he called them "old friends" and even enjoyed their singing (I made a detailed post about this here). Iroh aided Zhao in his assault against the Northern Water Tribe, he gives him a reasonable advice ("It's almost twilight, Admiral. As your military consultant, I must advise you to halt your attack. [Cuts to shot of the ship, sailing past a rising moon.] The waterbenders draw their power from the moon, and it is nearly full tonight. You should wait and resume the attack at daybreak. [Cuts back to Iroh and Zhao.]") Iroh is cool with his father, Firelord Azulon, who organized the Southern Raiders and established concentration camps; he remembers the man smiling at him in the LotFN book. And even when using the fandom's favorite word 'warning signs' since childhood, Iroh claimed he recognized destructive tendencies in Ozai but didn't deny that his brother still has the potential to turn out a decent person if he was guided and didn't suffer from favoritism under the hands of their father.
Did I mention the thief and the founder of the Red Lotus, Xai Bau? He teached the thief how to be truly intimidating if he wants to steal someone before he comes to understand his story. Iroh as well hoped one day the Red and White Lotus can work together. There was no shaming, insults or judgments. No righteous rage whatsoever.
Before his retirement Iroh dreamed of being Fire Lord, to conquer Ba Sing Se, admitted how fearsome he could've been in the eyes of the world, but one of the worst things Iroh done as a person is using the Secret of Firebending, life-giving qualities of fire, for conquest. Not only by this he earned his power boost, reputation and title (i.e. the Dragon of the West) but more importantly led his Nation imperialism and caused the death of many on the friend and the foe sides; he also was a bad influence on his 10-11 and 8-9 years old niece and nephew by literally laughing at burning down a "magnificent city" from the war front in a 600 days siege, while sending them gifts he took from the resistance forces (i.e. the dagger).
Only Lu Ten's death and the Spirit World what led Iroh to a rude awakening. But it wasn't enough. Again, Iroh didn't do anything to stop Ozai for years. He aided the FN war against the Northern Water Tribe; his only objection is for the moon spirit not to be killed, which is something anyone can understand. At the end of the show, the writers left Iroh where he was the most happy: profiting from the most privileged EK people who didn't even know there was a war going on of the very same city he tried to invade! Leaving Zuko, a teenager boy, who's clueless in politics to do the heavy lifting.
I see Iroh's repentance involved him only doing the bare minimum in stopping the war, he was mostly hedonistic and prioritize retirement over atonement. His most notable act of repentance is his determination not to repeat the same mistakes of not being a good father to Lu Ten (whom he led to his doom) and not being a good brother/guide to Ozai; to help him become a better version than the worst of himself. Note that he calls Ozai "brother" but never once called Azula a niece.
So, what does this has to do with Azula?
A. Azula is a special kind of evil which made her somehow worse than the mass murderers mercenary group who would throw in a party after destroying villages for shit and giggles, worse than the man who set up concentration camps, worse than the man who set on a propaganda bring out the worst in the Fire Nation, worse than the man whose pride and anger management can destroy a continent, worse than the man who established an organization to cause chaos in the world, worse than the man who purely led by his arrogance and ambition, worse than Iroh himself who literally used the true essence of firebending (ie life-giving qualities of fire) to kill people combined.
B. There is a different reason. The feelings he harbors for Azula aren't that of righteous rage; because we know that Iroh to a great measure is not a morally judgmental person.
I understand the menacing Azula vibes the show tried to depict and Iroh's wank and protection, but the audience shouldn't necessary get baited (i.e. Iroh isn't 'good' in the same sense Aang and Katara are; he's very morally grey). After examining Iroh's attitude toward the literal rest of the cast besides his own dark, shady side we can look for more reasonable reasons. Because if the case was simply Azula being "evil" then it's lacking and badly written, which is why many people, including myself, have questioned and rejected this logic.
Conclusion: Iroh dislikes Azula because she's not a good sister to Zuko, because she's an obstacle to Zuko's throne and because she shielded Zuko from Ozai's favor which results for Ozai to neglect Zuko in the process and this led to Zuko's misfortune.
(Iroh describes Azula as competitive and precise, which is true but he considered them as deadly and objectively bad qualities.)
Iroh's opinion can be biased more than based, because of his essential lack of relationship with Azula -- and considering the amount of time he spent in his Spirit World journey after Lu Ten's death (which wasn't specified in the show or its sources) it means he only spent little time in the Palace until Zuko's banishment (a point in time which Azula was probably at her meanest and competitive). He seems to was cool with little Azula to send her a doll (a gift anyone who had 2 minutes talk with Azula would known it's not the right/good gift), then after that she becomes only a threat in his eyes.
However, the interesting part is that Azula is not the only person whom Iroh gives this treatment, he did that with the Gaang and random people too. Yes, she received the most obvious and aggressive end of a passive-aggressive attitude which his love for Zuko leads, but she was not the only one who suffered from Iroh's favoritism:
I'd like to point that Book 1 Zuko especially kept hurting people repeatedly, you note that Iroh didn't stop Zuko from marching his ship into a populated village where Zuko almost killed a child, Iroh enforced a huge fireball by Zuko's side to stop Aang's escape (the Avatar), Zuko kidnapped and blackmailed Katara in front of him while he did nothing, he stands by while Zuko attacking Suki and set the village she tries to protect in fire, how Iroh was aware Zuko been stealing even things they don't need just because he can't stand the life of a commoner; yet his nephew was still his priority more than the amount of people who have their stuff stolen for no good reason. In LotFN book, Iroh outright downplayed Sokka's intelligence and claimed he needs Zuko to stay grounded, because Sokka is such a fool. Fans will make all the nitpicking over how Iroh secretly helped the Gaang but would blissfully ignore the elephant in the room that Iroh was constantly prioritizing Zuko's potential growth over other people's safety, which is why he's most of the time allowing it as long as it serves Zuko in the long run, especially if that person was Azula. So by keeping this consistency, Iroh believes that defeating Azula would be Zuko's passage into manhood is true to his character. He considers her Zuko's final enemy.
These are Iroh's goals and they were fulfilled: Zuko become good. Zuko doesn't have to do anything with his family, especially his sister. Zuko become Fire Lord and change his Nation. End story. Even in Legacy of the Fire Nation book, everything he said was clearly in regards Zuko's legacy, Zukoās wishes, Zuko's heroism, talent and intelligenceāZukoās everything.
Sources: ATLA TV show and the Legacy of the Fire Nation book (which written based on episodes as written, no additional content).
P.S someone's added a good naunce on this topic here.