When you do things warily, you're cautious and concerned. Behaving warily is like treating the situation with kid gloves.
There are a lot of things in life that should be approached warily — like bear cubs, for example. (Actually, they shouldn't be approached at all). Delivering bad news is done warily. People drive warily on ice, and they walk warily on ice, too. When you're acting warily — in other words, being wary — you're worried about what could happen, so you're careful and you take precautions.