What events from To Kill a Mockingbird make Atticus an irrational person?
I'm not sure I would call Atticus irrational at all, not in the truest sense of the word as "without the faculty of reason; deprived of reason." I might call him irregular with the meaning of "not conforming to established rules, customs, etiquette, morality," but that's not really the same. Now, from a racist perspective, Atticus no doubt looks very irrational for a number of reasons.
One- He knows from the outset that he will lose the Robinson case. He even tells his brother he will lose. And yet, instead of providing a token defense, he throws everything into winning a case which is unwinnable. If you take Atticus' actions on their face value, this seems illogical, but Atticus is not trying to win Tom's case in front of the jury. He's trying to set himself up for a good chance at appeal and he's trying to get people to recognize their biases, which many do.
Two-Atticus can't protect his family from racism, yet he constantly tries. The racist school system and even members of the town such as Mrs. Dubois bash Atticus, and yet he calmly accepts their hatred and tries to teach his children to approach race and human beings from a different perspective. From a racist point of view, this is irrational and confusing.
I would never say Atticus is irrational, but he does march to the beat of his own drummer, and his methods are neither conventional nor regular.
Atticus isn't irrational at all. He is, actually, for most of the book, infallible. However, at the very end, when Heck Tate speaks to him about Bob Ewell's death, Atticus believes, incorrectly, that Jem killed Bob Ewell somehow. In this, he is irrational in believing that Jem could have stabbed Bob Ewell when Bob Ewell is so much stronger than him, not to mention Bob Ewell broke Jem's arm.