In my opinion, the main irony in this story has to do with Aylmer and Aminadab. The irony is that Aminadab knows way better than Aylmer what is going to happen. This is ironic because Aylmer is supposed to be smarter than his assistant.
In the story, Aminadab is supposed to be pretty dumb. The description of Aminadab makes it clear that he is not really all that smart and Aylmer definitely treats him that way. Yet, from the beginning, Aminadab thinks that it is a bad idea to try to remove the birthmark.
As it turns out, Aylmer was wrong -- he was unable to remove the mark without killing Georgiana. So the intellectual man did not understand what the dumb man did. That is ironic.