He cannot stand brothers threatening their siblings, especially if they're the older brother. His philosophy is that older brothers are born first to protect the siblings that follow.
Do not insult the people he cares about and has sworn to protect. It doesn't matter if it's his dead mother, his school friends or his True Companions: don't mock them, don't threaten them, don't insult them and definitely do not rub their fates in Ichigo's face if they're the killer or kidnapper of said person.
Do NOT render him powerless to rescue people and then force him to listen to their dying screams. THAT will make him go ballistic.
NOT played for laughs. When Mayuri rubbed his quincy experiments, especially the torture of Souken, in Uryuu's face, Mayuri received more than he bargained for when Uryuu's response to explode right into a One-Winged Angel form, that not only turned him into a Walking Wasteland, but also one-shot both Mayuri and his bankai.
Orihime Inoue: Don't try to kill Tatsuki in front of her. To date it's the only example of her killing intent becoming powerful enough to one-shot the culprit to death.
Played for Laughs: Treating a girl improperly, such as attempting to kiss her will flip her out, especially if Orihime or, worse, herself are the targets of the amorous attentions. The teacher actually thought there was a demon in the classroom when it happened.
She will beat anyone who picks on Orihime to a pulp. As the second strongest female high schooler in Japan, she'll do a lot of damage, too.
Never do anything that will make her believe you have betrayed her trust. She will put your head through a window if you do.
Do not harm, insult or kill his lieutenant, Choujiro Sasakibe. It sent him into a long-term state of Tranquil Fury that eventually blew like a volcano in the most epic ignition of a Berserk Button in Bleach history. He's been called Old Man Genocide by the fandom ever since.
Gin Ichimaru: Cross him and he will spend decades plotting your death. Aizen's Rukongai experiments made a victim of Rangiku Matsumoto so Gin spent a hundred years plotting Aizen's destruction.
Izuru Kira: His War Is Hell personality makes him very quiet and calm even when fighting. However, bring up Gin, a man he was absolutely devoted to, and he'll sacrifice calm for a quick kill.
From the Arrancar Arc on, as far as his sister's health and safety is concerned, he's a Knight Templar Big Brother. He will kill for her.
The camaraderie of an ally is extremely important to him. Mess with the bonds between friends and allies, and he'll kill the fool who does it.
Shunsui Kyoraku: He is the Brilliant, but LazyChivalrous Pervert who would rather spend his days sleeping, drinking, or flirting. Unless the enemy is stupid enough to take out Heterosexual Life Partner Ukitake right in front of his eyes. Cue his laid-back personality switching to vicious, deadly serious mode.
Played for Laughs: He's sensitive about his youth, so he insists on being addressed as "Captain Hitsugaya". This is a problem where Hinamori is concerned as they grew up Like Brother and Sister and she still calls him childish nicknames such as Shirou-chan which she Will. Not. Stop. Doing.
Played for Laughs: He loathes his short height. When he's called a midget by the even shorter Hiyori, it tips him over the edge.
He has sworn to protect his best friend, Momo Hinamori. Seeing her injured will press his buttons. This is built up over the story until it leads to a catastrophic Roaring Rampage of Revenge when Aizen manipulates Hitsugaya into accidentally stabbing Hinamori.
Ikkaku Madarame: He's Bald of Awesome and knows it. However, if anyone claims he's bald rather than shaven, they can expect more trouble than it's worth.
Don't call him ugly. It's not worth the consequences.
The colour purple is the Berserk Button of his Empathic Weapon which is supposed to reflect the true nature of its wielder. Yumichika keeps calling his zanpakutou a fake nickname "Fuji Kujaku" (Wisteria-Purple Peacock) because it will refuse to give him its power if he does so. This allows him to fight with a fake shikai and sheer grit that mimics a melee-type zanpakutou because it's taboo in his division to possess a kidou-type zanpakutou.
Comparing Mayuri and Urahara together will send Mayuri into fits of rage and earn you threats of Cruel and Unusual Death for your trouble.
Abuse science for selfish or ignoble ends and Mayuri will kill you. He's a For Science! type of guy and that means he'll kill to protect his vision of it.
Ulquiorra: He cannot handle the idea of humans possessing the ability to fight on the level of the Espada or worse, surpass them. He went from emotionless to a raging monster inflicting a No-Holds-Barred Beatdown when Ichigo suggests their growing equality in battle means either Ichigo's becoming more hollow or Ulquiorra is becoming more human.
Do not call him ugly. It's not worth the consequences.
Riruka: She cannot handle being compared to a normal human or treated like one. She flips out when it happens.
Tsukishima is a cold, manipulative asshole who doesn't react with visible emotion beyond curiosity and amusement. But killing Ginjou flipped him so far over the edge he was able to function with a hole in his chest that was killing him. He died only after his rage and despair were spent.
Anime-only butons that don't ever crop up in the manga, include:
Ganju Shiba will get angry enough to drive a bucketload of customers out of his store if you mention his eyelashes.
Mention Aizen's betrayal and Hinamori flies into a rage.