If you spend $50 a day on food in NYC with a sales tax of ~9% we are talking about $4.50 a day. And $50 a day is high, if we are talking about people that are so price sensitive that they won't come to NYC at all without AirBnB.
Of the top 10 rooms listed in the AirBnB report, $100 a night seems to be the median price. With a hotel tax of ~6%, that should be $6 a day.
You can nab a real hotel room in NYC for $200 a night (with some work, even less). That yields ~$10 to the city a night in hotel fees.
It's hard to say without knowing a lot about price sensitivity whether or not your argument is true, but on the surface of it, I don't see it. Are there that many people that can buy a plane ticket, pay $50 a day in food, and $100 a night for AirBnB, but CAN'T pay $200 a night for a hotel?
I would imagine that more likely it changes the length of peoples stay. Maybe with AirBnB you can come for 6 days instead of 4 or something like that. But if AirBnB currently doesn't pay any taxes, I'm don't see how 6 days of sale tax instead of 4 makes up for the lost hotel tax.