"straight edge drinking game" is a humorous expression that implies that a certain event or series of events will not happen, or will happen with very little regularity.
It's equivalent to saying "that will never happen", "no chance". "not in a million years" or other idiomatic expressions.
Example 1:
Joe: Hey, let's drink every time Bush says something smart.
Paula Abdul said it better than I could with the chorus : "Straight up now tell me,
Do you really want to love me forever? Oh, oh, oh...
Or, am I caught in a hit and run?
Straight up now tell me,
Is it gonna be you and me together? Oh, oh, oh...
Are you just having fun?"