You can keep green mana in your mana pool indefinitely while Omnath is on the battlefield. That means if you add a green mana to your mana pool 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 continue to empty from your mana pool as each step and phase ends.

If a green mana you add to your mana pool 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 whatever green mana is in your mana pool before it too empties as normal. There is no penalty associated with this other than the loss of the mana.