Dr. Heidegger's Experiment
The rose anticipates what happens to the guests. Dr. Heidegger first uses it to demonstrate the rejuvenating power of the elixir; and later it withers right before the same thing happens to the old guests. If you believe that the elixir is nothing more than alcohol, then the rose is a key part of the dramatic show that Heidegger puts on for his guests in order to convince them that they are in fact growing young and then old again.
You also have to consider the importance of the rose to Dr. Heidegger. He reveals that his fiancée, Silvia, gave him the rose to wear to his wedding. Now that she's dead, of course, the rose has great sentimental value. "I love it as well thus," the doctor says to the withered rose "as in its dewy freshness" (47). This is another example of Heidegger's wisdom in contrast to the folly of his guests. While they place importance solely on youth and beauty, Heidegger's emotions, like his character, are not so shallow.