Rollie Fingers lost his chance to play for the Cincinnati Reds when he told the team yesterday that he would not shave off his mustache to conform with the Reds' policy that players not display facial hair, a Reds executive said. The Reds had expressed interest in having Fingers, a veteran reliever known for his mound effectiveness and handlebar mustache, join the team.

But Fingers and General Manager Bill Bergesch were unable to reach agreement in a telephone conversation yesterday on the issue of shaving off the mustache, and Bergesch broke off the Reds' interest in Fingers. After their conversation, Bergesch told reporters: ''Fingers started off by asking me if what he had heard and read was true, that is, would we require as a condition for trying out with the Reds, his removal of his handlebar mustache? I told him it was Cincinnati policy and would have to be uniformly applied.'' Bergesch said that Fingers' reply was final. He quoted Fingers as saying: ''The mustache is my trademark, and it has been for 15 years. I am not about to shave it off just to play baseball.''