What do you do with the gum in Nancy Drew?

Answer 1

If you're referring to the computer game, "The Final Scene", where the gum is found on a movie theater chair [still sticky], you take the magic wand from Brady's closet, and stick the gum to the end of it. This will come in handy when you need to catch a gear off the edge of one of the balcony's in the theater.

If you're referring to the computer game, "Lights, Camera, Curses", you combined the gum with with the excess coffee in the coffee mug. Then you connect that new sticky substance to the end of the pencil and use that to get the key that is stuck in the back of the vent.