The most ironic scene in this play comes in Act III scene 4, which is when Hamlet is with his mother and he hears Polonius shouting, obviously fearing that he is going to visit some harm on Gertrude. Hamlet kills this hidden character thinking he is Claudius, only to discover that it is actually Polonius. Note what he says when he discovers the identity of this person:
Thou wretched, rash, intruding fool, farewell.
I took thee for thy better. Take thy fortune.
Thou find’st to be too busy is some danger.
What is ironic about this is that Hamlet thought he was killing Claudius, the man he needs to get his revenge on. Instead, he has only killed a "rash, intruding fool" who is in many ways a figure of fun in the play. Also, what adds to the irony is that know Hamlet has just made the parallel between himself and Laertes that much stronger. Both have now had their fathers killed by somebody and both therefore have a right to revenge. Hamlet, through this act of killing, has sown the seeds of both his own destruction and the destruction of Ophelia. Hearing of her father's death, she is driven into madness and kills herself, and when Laertes finds out about what has happened, he returns swiftly to Elsinore to get his revenge on Hamlet, without any of the procrastination that Hamlet himself has displayed. The scene therefore is incredibly ironic, not just because of the mistaken identity, but because of the unintended and unforeseen consequences of that mistaken identity.