Ticklin' the Ivories
There's only one central location in Casablanca—Rick's. And right in the middle of Rick's is Sam's piano. When you also consider the fact that the piano is where the letters of transit are hidden—the one thing on which the entire plot hinges—you can probably bet that this thing is important.
The piano also provides a link to Rick's and Ilsa's past. It's where Ilsa first goes when she enters the café; she asks Sam to play for a song that used to be "their" song, a song that Rick has refused to let Sam play because it just hurts too much to hear it. Even more painful, it's led generations of moviegoers to misquote "Play it again, Sam."
If we think about the broader meaning of the piano, we obviously have to think about music. One of the most essential, stirring scenes in the entire movie is when the German officers are singing their anthem, and then Laszlo swoops in and gets the orchestra to start playing France's anthem. It's representative of the whole political struggle that's embedded in this story, and there's probably nothing like music that could drive this point home any more, or bring a wetter, soppier tear to our eye.
But even on a more abstract level, music is the expression of the soul; it steps up to the plate when words are insufficient. And there are so many times in Casablanca where, despite all the brilliant dialogue, something can only be said with a look, or a touch, or a kiss. Which is why, when you hear two people say they're going to make sweet music together, they're, um, not always talking about a couple of dueling banjos.