The Rose
Will you accept this rose?
Symbols don't get more symbolic than the enchanted rose, which sits under glass in the Beast's Unhappy Place and generally acts as the ticking clock on his cure. It will bloom until his 21st year, giving him a timetable to find a nice girl and prove to her that he isn't a giant jerk before he gets stuck in pseudo-werewolf mode forever.
Roses, as you may have suspected, are chock full of symbolic meaning to begin with. They represent love, for starters, which obviously connects to the whole pointof this movie. In the traditional fairy tale, the wilting of the rose symbolizes Beauty leaving the Beast, making us wonder if their love is going to last or not. Here, it shifts a little bit to be a little more indirect: symbolizing the Beast's chances at love rather than the shared loved between the couple.
Roses also stand in for death, especially when they're dying like this one. They hide thorns under their leaves, which can stick you if you get too close; their passing often symbolizes the passing of everything worthwhile in life. That's a pretty close approximation of what the Beast feels: if he can't break the curse, then he might as well be dead. (Witness his indifference to Gaston slaughtering him in the climax before Belle arrives.)
Finally, roses are also symbols of mystery: opening slowly to reveal their beauty only after you take the time to get to know them. That's in keeping with the whole romance thing, but in this case, the rose has already bloomed and begun to wilt. For Belle, it represents a different sort of mystery: the mystery surrounding the Beast and how he came to be that way. That's the vibe we get when she first finds the rose in the Beast's "forbidden" rooms. While the mystery gets answered fairly quickly, that doesn't change what it represents for her.
It's a lot for one lowly little flower to take on. Thankfully, this one seems up to the task and does its job right until the moment the final petal falls.