For those that don't know, Mr. Brando won Best Actor for his role in The Godfather at the 1973 Academy Awards. He was not present at the event and instead sent an Apache woman named Sacheen Littlefeather to announce that he was refusing to accept the award as a protest against "treatment of American Indians by the film industry and on television" and also because of the 1973 incident at Wounded Knee, SD, that resulted in several fatalities. You can read about that incident here: http://en.wikipedia.org/wiki/Wounded_Knee_incident and Ms. Littlefeather's appearance at the Oscars can be seen here: http://youtu.be/2QUacU0I4yU
The answer is in dispute.
According to the Academy of Motion Picture Arts and Sciences, Roger Moore (who was to present Brando with the Best Actor award) took it home with him that night but later returned it to the Academy. It was then relabeled and given to Charlie Chaplin to replace a damaged award.
Comedian, actor, and talent agent Marty Ingels asserts that one of his clients (who he refuses to identify) walked away with the award that night and still has it to this day.
See this 2004 article in Entertainment Weekly.
Was there backlash against Brando as a result? The speech that Sacheen gave seemed pretty respectful.