I am struggling to come up with an example of dramatic irony in this excellent short story. It is certainly a brilliant example of situational irony, but I don't think we can identify any instance of dramatic irony. Let us remind ourselves that dramatic irony occurs when the audience, and sometimes one or more characters, knows a vital piece of information that another character or group of characters does not. The classic example is of course in Romeo and Juliet when we know that Juliet is not dead, and that she is about to wake up, but Romeo doesn't, and kills himself just before she wakes up.
The closest we can perhaps come to dramatic irony is the ending, when we as the audience and Mr. White recognise that what is knocking on the door is not going to be their son as they remember him, but a hideous, terrifying corpse that will make them wish that they had never meddled with their fate. The way in which the mother responds to this knocking shows that she does not understand this, however:
"You're afraid of your own son," she cried struggling. "Let me go. I'm coming Herbert; I'm coming."
Mr. White's success in getting to the monkey's paw before his wife manages to open the door shows this dramatic irony at work, but I don't necessarily think this is a particularly good example of dramatic irony. This story is a much better example of situational irony.