How long does burning tree emissary's mana that's is given basically float around for use? Does it just stay there until used like 5 turns later or does it disappear right after your turn? {Burning-Tree Emissary}
It goes bye bye at the end of the phase/step. To further extent all mana in your mana pool does.
From the Comprehensive Rules:
106.4. When an effect produces mana, that mana goes into a player's mana pool. From there, it can be used to pay costs immediately, or it can stay in the player's mana pool. Each player's mana pool empties at the end of each step and phase.