What do I do with items I don't want that come out of the Vault?

You can easily sell items while playing the Vault of Riches using the Sell button in the lower-right. The fastest way is to drag an item over the button to sell it. You can also select an item, then tap the button.

You should sell any items you don't want before exiting the Vault of Riches so that they don't clutter your Garden.