Although I wouldn't quite call that one a "heartbreaker" (probably because I was more pissed off than disappointed), that New England game has to be way up there.
Another one was the 2001 (?) game at home against the Pats. We sucked that year, but we played them tough and deserved to win it. But the mysterious "ball lying inbounds on the ground touching a frigging unconscious player is considered out of bounds" rule ruined it for us.