There's some situational irony at play in "The Fly," especially when the boss emotionally falls apart after thinking about his son's death. Although the boss likes to think that he's in a better state of mind than Mr. Woodifield, the opposite ends up being true—whereas the doddering old Woodifield seems to have processed his own son's death, the strong and powerful boss goes into an emotional tailspin at the mere mention of his son. The irony, then, is that he's actually much more troubled and out of sorts than Woodifield, even though he seems much more composed than his old friend.
In the beginning of the story, Mr. Woodifield is presented as absent-minded and feeble. The implication is that the boss sees Woodifield as somewhat pathetic. When Woodifield has trouble remembering something he wanted to tell the boss, the boss privately thinks about how Woodifield is a "poor old chap" who's slowly losing his mind. However, Woodifield eventually remembers what he wanted to say, explaining that his daughters recently paid a visit to his son's grave and happened to also see the boss's son's grave. From this point on, the dynamics in the story begin to reverse, as it becomes clear that Woodifield is perfectly capable of facing loss head-on, while the boss is completely unable to hold himself together after even the slightest mention of his dead son.
What's ironic, though, isn't just that the boss proves himself to be emotionally weaker and more sensitive than Woodifield, but that he also becomes more forgetful than Woodifield. The boss looks down on Woodifield's failing memory in the beginning of the story, but when he finishes torturing the fly (as a way of distracting himself from his tumultuous emotions), he himself can't even remember what was troubling him. Of course, this is most likely the main reason he tortured the fly in the first place: to keep himself from thinking too much about his dead son. Still, though, it's ironic that the story ends with him wracking his brain in an attempt to remember what he was thinking about, since this kind of absentmindedness is precisely what led him to think disparagingly about Woodifield as a "poor old chap." In the end, then, it seems that the boss is the pitiable one.