In the obscurest corner of the room stood a tall and narrow oaken closet, with its door ajar, within which doubtfully appeared a skeleton. (3)
You've probably heard the metaphor of the skeleton in the closet before. The phrase refers to some sort of dark secret a person is keeping hidden. It's both hilarious and sinister that Dr. Heidegger literally has a skeleton in his closet. Not only is this a perfect example of the narrator's attitude and technique (see "Writing Style" and "Tone"), but it does wonders for the complexity of Dr. Heidegger's character. What is he hiding in his past, exactly? What is his metaphorical skeleton in the closet?
This brings us rather quickly to the issue of Heidegger's dead fiancée, which is one heck of a closeted skeleton (though, we soon see, not so carefully closeted). We see a large portrait of a young lady, and get a brief explanation from the narrator: "Dr. Heidegger had been on the point of marriage with this young lady; but, being afflicted with some slight disorder, she had swallowed one of her lover's prescriptions, and died on the bridal evening" (3).
This is not much of an explanation, as explanations go. It's incredibly ambiguous; did Dr. Heidegger prescribe the wrong drug for his fiancée? If so, was this accidental, or intentional? Is his medical incompetence to blame for her death? Or is there something even darker going on here? Another possibility is that his fiancée (we later find out her name was Sylvia) took one of the drugs in Heidegger's stores without consulting him first, accidentally causing her own death. Suicide seems less likely here, but again, because of the ambiguity of this line, not outside the realm of possibility. (Her "slight disorder" could have been psychological for all we know.)
The point is that these two items – the skeleton in the closet and the portrait on the wall – raise more questions about Dr. Heidegger than they answer. The real nature of his character – in particular whether he is benevolent or sinister – is one of the story's biggest questions.