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3.
abandon something or someone
Dates from the early 1970's, derived from waving goodbye, a similar hand motion to polishing something.
This project is no longer relevant. Time to shine it on.
by KitCur October 15, 2008
 
1.
From back in the '70s, meaning to "forget it" or "never mind"

"You wanna go out with me?"
"Well, I'm kinda busy..."
"Okay, shine it on."
by Louisa D November 11, 2006
 
2.
to act like it doesn't bother you
Ricky's behavior was uncalled for, but I knew it was best that I just shine it on.
by Joey Low January 25, 2008