to shine someone on, as used in Terminator 2 by John Connor (Edward Furlong) he goes on to say that "it's hasta la vista, baby", which in a literal translation says "until the view, baby" and more loosely means "until i see you again", baby. So what he's getting at is this: to give someone the impression that you're attracted or into them, but they will have to wait and see, you say "Hasta la vista, baby". So, by using the spanish farewell in a conversation in English he's acting cool, like. Right? Thus, he is "shining" them on.
So a similar thing would be to string someone along or to lead them on.