Other mistake: When Glen Holland hears that John Lennon was killed, he is sitting at his piano and outside the window it is sunny and leaves are on the trees. As he walks home, it is overcast, snow is on the ground, and the leaves are off the trees. Since Lennon was killed in December of 1980, the latter "weather" would be correct.
Revealing mistake: Anytime the camera is on Richard Dreyfuss' hands while he's playing the piano, it is clear that he is not actually playing; his hands are frequently in the wrong part of the keyboard for the sound being produced. At other times, when his hands are moving down the keyboard, the pitch we hear is going up, or vice-versa.
Other mistake: At the beginning of one of the school years, there is a sign on the front of the school that says, "Welcome back Class of 1980". A few months later, in December of that same school year, John Lennon is murdered. Since that class is graduating the following spring, the sign should have read "Welcome back Class of 1981", not 1980.
Continuity mistake: At the end, when Mr. Holland is conducting his symphony, there is a closeup of Iris lifting her camera up to take a picture. The shot cuts away to show Mr. Holland. You can see Iris and Cole in the background on the lower right of the screen and they are holding hands with no camera in sight.