When John Barrowman was in Sunset Boulevard he chewed gum, but it was due to having forgotten to take it out during rehearsal(I think they were in the midst of a change of the actresses playing Norma from Buckley to Paige: I could be wrong) and he noticed it afterwards and he was about to spit it out when the director said to keep it because it worked the with the scene. I don't remember what scene they said it was, it was a funny moment of forgetting it was there but it worked quite well with the scene apparently.
As for my experience my teachers have always said to take out gum. And most of the time I would remember. Admittedly sometimes I'll forget during rehearsal, but most of the time the director won't notice. Never have I chewed gum onstage. Sometimes like everyone else has said I think it's a choice for the character/scene and sometimes it works other times it doesn't. It just depends.
"The weight of this sad time we must obey, Speak what we feel, not what we ought to say. The oldest hath borne most; we that are young Shall never see so much, nor live so long"-Edgar in King Lear