Hester's Refusal (Situational Irony)
Hester's insistence on staying in the Bog of Cats and her repetitive refusal to leave represents a kind of situational irony. Even after she has been told she must leave, has sold away her own house, and has been told by a prophet that she will die by the end of the day if she stays, Hester does not budge and insists on staying there. This creates an ironic situation in which, even though she and the audience know that she will meet a tragic fate, she refuses to take that seriously.
Burning the farm (Dramatic Irony)
While everyone attends Carthage's wedding, Hester lights the farm on fire. As it blazes, we the audience know what is going on, but none of the other characters do. Because of the destructiveness of the act, we know that the other characters, especially Carthage, will not be pleased. This disparity between what the audience knows and what the other characters know creates an instance of dramatic irony.
Hester killing Josie (Dramatic and Situational Irony)
Hester's murder of her own daughter is ironic on multiple levels. On one level, it is ironic that she would kill her daughter, the one person she loves the most and the one whom she most fears losing. In a fit of madness, she believes that by killing her daughter she will be able to be with her forever in the world of the dead.
On another level, this moment is an instance of dramatic irony, because the audience witnesses the murder before any of the other characters do. There is a short period of time in which we know about Hester's horrific act before anyone else does.
Caroline and Hester were once friends (Situational Irony)
Hester deeply resents and hates Caroline Cassidy, because she is the new woman that Carthage is marrying. She hates that Caroline is inheriting her old life, and wants to make sure that Caroline knows that she is irreplaceable. However, later on in the play, we learn that Hester and Caroline were once friends, as Hester used to babysit Caroline, and acted as a sisterly figure after Caroline's mother died. Thus, it is ironic that they are now such enemies, since they actually share a lot of history.