I would say that the situational irony in this story is used to emphasize the main point of the story. I think that the main point is that you can't really change fate or destiny -- if you try, it might end up backfiring on you.
In the story, the White family tries to manipulate their fate by using the monkey's paw. This is where the situational irony comes in. Both times that they wish for something, they do not get it in the way that we think that they might. We expect them to get the 200 pounds in some lottery or something but instead they get it from their son dying, for example.
So the situational irony is meant to emphasize the main point of the story.