If unspent mana has any restrictions or riders associated with it (for example, if it was produced by White Lotus Hideout), those restrictions or riders will remain associated with that mana when it becomes red.

As long as Ozai, the Phoenix King is on the battlefield, you'll keep unspent mana as steps and phases end (although it will become red). This means you can add mana and spend it during a future step, phase, or turn. Once Ozai leaves the battlefield, you have until the end of the current step or phase to use the mana before it disappears.