A fruit juice stain will probably work. Don't call it a finish though, it's a stain, not a finish. You need to apply a finish on top of the stain, for instance tung oil like you mentioned.
Most fruit stains are not colourfast at all. They will darken first, to a shade of brown probably. After that, they'll fade over time. Don't expect a longlasting red colour from the strawberries and a blue color from the blueberries. All will turn brown quickly.