No, treasure cards stay in play. You may view the process as follows:
- During your action phase, you play any action cards (or treasure cards, as well with the help of the Black Market) that generate Coins.
- You play the Storyteller; you gain +1 Action and +$1.
- You may play now up to 3 treasure cards from your hand.
- Any total money that you have generated and have available so far (as you might have spent some with the Black Market), is lost; you draw that many cards.
- Any subsequent money you generate is not lost and may be spent in the Buy Phase.
While the Storyteller card explicitly mentions that you have to "pay all your coins", it does not mention anything about any card leaving play. A card only leaves play during the Clean-Up phase (base game rulebook) or when a card effect explicitly says so (e.g., the Mint from the Prosperity expansion, or the Duration cards).
Furthermore, the Adventures rulebook (p.15) makes it crystal clear that "spent" treasure cards stay in play:
You can track that the Treasures have been "spent" by putting them under the Storyteller.
However, if for some reason you have $5 and two buys, it is perfectly legal to buy a Mint for $5, trash all your treasure cards in play because the Mint says so, and then buy the Alms event, gaining a card costing up to $4. But not with the Storyteller.