And the answer is: nobody. That line isn’t in the movie. We get the full scoop from the website The Phrase Finder:
This is well-known as one of the most widely misquoted lines from films. The actual line in the film is ‘Play it, Sam’. Something approaching ‘Play it again, Sam’ is first said in the film by Ilsa Lund (Ingrid Bergman) in an exchange with the piano player ‘Sam’ (Dooley Wilson):