Think about the function and type of this story. What is the purpose and how is it presented? Clearly, this is a richly allegorical tale to show the futility of wishing for eternal youth and the way that humans inevitably repeat their folly. The purpose of this story therefore has a darkly comic slant when we witness, along with Dr. Heidegger, the four guests swear that they would never repeat the mistakes they committed in their youth only to see them fall into exactly the same kind of vices that ruined them in the first place. Note the way that the guests respond to Dr. Heidegger's recommendation that they take steps to ensure they don't repeat their mistakes:

The doctor's four venerable friends made him no answer, except by a feeble and tremulous laugh; so very ridiculous was the idea that, knowing how closely repentance treads behind the steps of error, they should ever go astray again.

How darkly comic, therefore, that within minutes we see each of them doing precisely this, and then not only this, but when old age reasserts itself, they are blind to what Dr. Heidegger has learnt about the value of age and the benefits of the wisdom that comes with it. This is where the dark humour of the text lies: 

But the doctor's four friends had taught no such lesson to themselves. They resolved forthwith to make a pilgrimage to Florida, and quaff at morning, noon, and night, from the Fountain of Youth.

Human folly is exposed and mocked. The humour is of course dark because by laughing at the guests we unwittingly condemn our own tendency to repeat our folly and the way that we, too, have failed to learn the lesson of this story.