If I am collecting unemployment, do I have to claim my earnings from Rover? What if I leave my earnings in my Rover account until I start working again?
Each state has different requirements, but in general you are considered an independent contractor so you are required to report your Rover earnings to your unemployment office when you are paid by Rover. it does not matter if you withdraw the funds are not, Rover technically paid you and you 'earned' the money during the time of the stay. I would recommend consulting your local unemployment office for specifics. I know in Texas when I was drawing unemployment I was allowed to earn 25% of my unemployment check before they started decreasing the amount of their payment.
When the company paid you (i.e., put it into your account) is when you earned it and would have to report it. Just because you've left it in a noninterest-bearing account at Rover has nothing to do with anything.
I am not an accountant, so I definitely suggest reaching out to someone that has legal tax knowledge but as far as I am aware, if you haven't withdrawn anything from your Rover account in 2016, you haven't actually received the income, so you don't have to claim it. That being said, when you do withdraw it, you will have to claim everything you withdraw in the year you withdraw it. But once again, I am not an accountant or tax person, so please double check this answer.