OED does credit the 1969 movie with popularizing the term but the first cite for this sense is from a song released a year earlier. (Actually I think it’s the song they use in the movie.)
1968 ‘Fraternity of Man’ Don’t bogart Me (transcription of song) in http://www.stlyrics.com (O.E.D. archive) , Don’t bogart that joint, my friend Pass it over to me.
There is also a prior sense for the word.
1. trans. U.S. (esp. in African-American usage). To force, coerce; to bully, intimidate.
1966 Current Slang (Univ. S. Dakota) 1 ii. 1 Bogart, to injure or hurt, or to protect at the cost of violence.—High school male, Negro, Mid-Atlantic Coast (Washington, D.C.).—Them ‘bama chukkers better not bogart us no more.