Answer:
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.
