Michael, Teak oil is meant to soak into the wood, not dry on top of it. You shouldn't be worried about drips or anything. Let it soak for 12 hours or so, remove from oil, and wipe dry. Then a day or two to cure and you're done.
LifordCC, like I said above, don't worry about drips or any of that with Teak oil. Just drop it in for a 12 hour soak, remove, and wipe excess off. You shouldn't have to thin it, I never have.