A Series Of Unfortunate Events Season 2: 10 Miserable Fan Theories You Don't Want To See
6. His Cough Is A Metaphor, Or Another Literary Reference
Pivoting away from the notion of Mr Poe being involved in either good or bad capacity with the V.F.D., but still assuming that is cough isn't just a cough (now there's a plot twist), and there are still a couple of other possibilities thrown up (turns out people are really interested in his coughing).
One is that it's a sign of his incompetence; that the fact he can't take care of himself is a reflection or sign that he is not fit to be the person looking after the Baudelaires.
Alternatively (and personally I'd say this is the best theory), it could be yet another literary reference. The books and subsequently show are littered with them, and this would be, like his name, another reference to Edgar Allan Poe. The writer's wife died of tuberculosis, and one of the symptoms of that disease is severe coughing.