Simply put, dramatic irony is when a person makes a harmless remark and someone else who hears it knows something that makes the remark have a different, and usually unpleasant, meaning. For instance, if you are in a restaurant and said out loud, "I can't wait to eat the veal marsala I ordered," and there were people around who knew that the veil marsala was poisoned and that you would die as soon as you took a bite, your situation would be one of dramatic irony.
Uncle Monty: "I promise that if you take time to learn the facts, no harm will come to you here in the Reptile Room."
As you and I listen to Uncle Monty tell the three Baudelaire orphans that no harm will ever come to them in the Reptile Room, we should be experiencing the strange feeling that accompanies the arrival of dramatic irony. This feeling is not unlike the sinking in one's stomach when one is in an elevator that suddenly goes down, or when you are snug in bed and your closet door suddenly creaks open to reveal the person who has been hiding there. For no matter how safe and happy the three children felt, no matter how comforting Uncle Monty's words were, you and I know that soon Uncle Monty will be dead and the Baudelaires will be miserable once again.