If you're going to count E.T. as mute, then you should count Quasimodo in
The Hunchback of Notre Dame, at least in some versions. I guess it's harder to tell if a character is a true mute if you're watching a silent movie, but the character of Quasimodo is basically played with body language and makeup, especially by Lon Chaney in the silent 1923 version and Charles Laughton in the classic 1939 version. True, this character can speak but only haltingly and under extreme emotional distress.
I can think of at least three actresses who, off the top of my head, won Oscars for playing mute characters: Jane Wyman in
Johnny Belinda (1948) who plays the rape victim title character, Patty Duke as Helen Keller in
The Miracle Worker (1962), and Holly Hunter in the exotic
The Piano.
In
Mute Witness (1993), Marina Zudina plays a mute makeup artist on a low budget slasher flick who witnesses the filming of a real snuff film and spends the rest of the movie in mortal danger from the killers and their boss.
There are dozens of other significant characters who are either really mute or playing mute, and there are hundreds of other supporting characters in hundreds of other films. I'm sure somebody else might want to add some before I hog the entire subject, at least as long as this is a legitimate request. All these films are easy to find from various sources.