How long does mana from Burning-Tree Emissary stay around?
The mana created by Burning-Tree Emissary is no different than then mana created by Forest and Mountain cards. Keep in mind the ability of a Forest reads "{T}: Add {G} to your mana pool."

As usual, it remains there until used or until the current step or phase ends.


  500.4. When a step or phase ends, any unused mana left in a player’s mana pool empties. This turn-based action doesn’t use the stack.