Bernard's Theories about Byron and Chater
There is a tremendous amount of dramatic irony when the audience realizes that Bernard's theories about Byron and Chater are untrue, just as he is publishing and promoting them. We start to see the true events of the story coming together as Bernard barrels towards his conclusion. For example, we know why Chater, Mrs. Chater, Byron, and the Captain left long before Bernard realizes that Chater was not shot in a duel.
Proof
It is ironic when Bernard chides Hannah for saying she wants proof of the handwriting being Byron's, when he has little proof himself of the things he argues.
"Et in Arcardia ego"
Socratic irony is present when Thomasina plays dumb in the discussion with her mother about the meaning of "Et in Arcadia ego." Her mother translates it incorrectly and Thomasina knows it, but chooses to allow her mother to persist in her delusion. There is also a tragic sense of irony in the fact that Thomasina ends up dying.
Relationships and Research
There is irony on a grand scale in the characters pursuing their relationships and research when the universe will grow cold and render all meaningless. Essentially, none of what they are doing matters - personalities or penicillin - because the universe will die. However, Stoppard does not counsel us to despair but rather to take pleasure in life, love, and the pursuit of knowledge.
