Will oil drip when saw isn't running? Empty oil tank between uses?

Mine weeps oil when it is stored. I store it with the shaft horizontal and the tank up. It's probably weeping out the same hole that is used to oil the chain.

The oil won’t drip out unless you are storing it upside down, for instance, when using the hanger device. If this is how you plans on storing the unit then you should drain the oil before putting it away.