The big question you find yourself asking as you read "Dr. Heidegger's Experiment" is whether or not the elixir he gives his guests is actually water from the Fountain of Youth that turns his guests young again; or whether it's just an illusion, a trick that Heidegger has cooked up in order to prove his point. You can make a case for either one.
There are enough mystical and spooky items detailed for us in the third paragraph to place the reader firmly in the realm of the supernatural. The narrator sets us up for other-worldly events before the hinge of the plot even unfolds. Heidegger has a book of magic, a bust that speaks, a skeleton in his closet, a portrait that can move, and a mirror that holds the faces of all his dead patients. This is not realism – it's fantasy, and so we should accept that the elixir is in fact fantastical.
The rose is great evidence for this as well; the guests haven't had anything to drink yet, booze or not, so they are in their right mind when Heidegger transforms the rose before their eyes. The actual transformation of the guests is equally explicit:
Even while the draught was passing down their throats, it seemed to have wrought a change on their whole systems. Their eyes grew clear and bright; a dark shade deepened among their silvery locks, they sat around the table, three gentlemen of middle age, and a woman, hardly beyond her buxom prime. (26)
Later, after the vase is broken, the rejuvenated butterfly reinforces our belief that the elixir is legitimate.
The narrator is careful to hint at the real nature of the elixir – it's not water from the Fountain of Youth; it's just alcohol. The sparkling bubbles on the brim immediately make us think of champagne, and this idea is reinforced by reference to wine over the next few pages of text. You'll also notice that the narrator never explicitly says the guests indeed are young again – he only says that they seem to grow young again. Take a look at the following passages (the bold sections are our emphasis):
The point of "Dr. Heidegger's Experiment" isn't science or superstition. Hawthorne isn't interested in the physics of growing young again, or the location of the Fountain of Youth, or the details of how Dr. Heidegger managed to procure the liquid, or any other trivial matters. He is interested in moral philosophy. The heart of the story lies in two key passages spoken by Dr. Heidegger:
"Before you drink, my respectable old friends […], it would be well that, with the experience of a lifetime to direct you, you should draw up a few general rules for your guidance, in passing a second time through the perils of youth. Think what a sin and shame it would be, if, with your peculiar advantages, you should not become patterns of virtue and wisdom to all the young people of the age!'' (19)
"Yes, friends, ye are old again,'' said Dr. Heidegger, "and lo! the Water of Youth is all lavished on the ground. Well – I bemoan it not; for if the fountain gushed at my very doorstep, I would not stoop to bathe my lips in it – no, though its delirium were for years instead of moments. Such is the lesson ye have taught me!'' (51)
As we discuss in "What's Up with the Title?", Dr. Heidegger's experiment tests and then proves a hypothesis: if man were allowed a second chance at youth, he will make the same mistakes again. One could test this hypothesis by procuring water from the Fountain of Youth, or by pretending to have procured water from the Fountain of Youth. But the hypothesis is proven either way. It ultimately doesn't matter whether or not the elixir is real.