You can keep unspent green mana indefinitely while Omnath is on the battlefield. That means if you add a green mana during one step or phase, you can spend it during a later step or phase, or even a later turn. Other types of mana will be lost as normal as each step and phase ends.

If a green mana you produce has certain restrictions or riders associated with it (for example, if it was produced by Ancient Ziggurat), they'll apply to that mana no matter when you spend it.

Once Omnath leaves the battlefield, you have until the end of the current step or phase to spend your unspent green mana before it too is lost as normal. There is no penalty associated with this other than the loss of the mana.