The Phantom of the Opera, when looked at closely, proves to have a rather idiotic plot. Why does Christine, after finding out that the Phantom is not the Angel of Music, fall for it again? Why doesn't Raoul just kill the Phantom when they're in the graveyard? Why don't the owners of the opera house ever investigate this "ghost" to whom they have been paying 20,000 francs per month? For that matter, why don't they call in the police force when said "ghost" not only violently kills a stage hand but also causes the chandelier to crash into the stage, nearly killing more people and costing who knows how much to replace?
Thankfully averted by the book on all counts.
Love Never Dies is even worse. For starters, nobody thought to tell Gustave "Listen, that guy in the mask? He's very, very dangerous, he's killed two people that we know about, he tried to kill your father, and if he gets his hands on you he'll hurt you or use you to hurt your parents. So stay close to us at all times and whatever you do, don't go wandering off with the creepy carnival folk, okay?"
In Othello, Desdemona probably wouldn't have died if Othello had just flat-out asked her "Were you having an affair with Cassio?" or waiting to point fingers until after Iago brings the "occular proof" that he had asked for. (Assuming Othello would have believed Desdemona.)
Shakespeare's The Comedy of Errors. Antipholus of Syracuse has been looking for his long-lost twin brother, and comes to a town where everyone seems to know him, including someone claiming to be his wife. Somehow, neither he nor anyone else manages to come to the obvious conclusion that this is where his twin has been living. Same with Plautus's The Menaechmi, upon which The Comedy of Errors is based.
Much Ado About Nothing. Claudio and Don Pedro already know that John is not a nice guy. And as if that weren't enough, Claudio gets taken in by John's claim that Pedro courted Hero before being disabused of the idea. So Claudio and Pedro know that John is trying to spread rumors to break Claudio and Hero up - why do they fall for the plot again?
In 13, everyone seems to be holding the Idiot Ball because they believed Lucy during "it can't be true".
In Clare Boothe's preface to her play The Women, she notes that if Mary, the principal character of her play, were a reasonably intelligent woman, she would quickly have found a different play to be in.
The opera la Sonnambula, by Bellini. Sure, she's in a compromising position... but it's completely out of character for her.