arch and Toady,
No, the kid (Billy Redden) did not play the banjo, but neither was it a sound over, according to this excerpt...
Billy Redden
This image is iconic in as much as it is rooted in American folklore, it triggers fears of loss and hopelessness in people. Billy Redden was not actually acting, he was the real thing and was not sufficiently expert in faking the playing of the infamous music sequence convincingly enough; ultimately another youngster was hidden behind his chair, with the task of providing the hand movements on the guitar.
Billy Redden is still alive and has made a second film with Tim Burton, Big Fish
...which I got from this site.