Concerning the Vitali Covering Theorem, what is the significance of closed balls in the hypothesis? In particular wouldn't open balls also work?

Assuming that you are considering the Vitali covering theorem for Radon measures in $\mathbb{R}^n$ the answer is that open balls don't work. I'll leave the task of finding a counterexample (for example already in $\mathbb{R}$) to you, as I remember it being a homework assignment on our real analysis course. (Notice that open balls work trivially for any measure that gives zero measure to the sphere.) 

