Please note that this FAQ answer is not legal advice and it is not comprehensive.
City of Seattle
If you are running an Airbnb in Seattle, registering with the City of Seattle is required. Because you are generating income in Seattle, the City considers you a business. Airbnb has a great help section about registering in Seattle, here.
State of Washington
If you make above $12,000 per year, you'll also be required to register with the State of Washington.
Cities Beyond Seattle
If you're operating an Airbnb beyond the City of Seattle, it's a good idea to check local regulations, too.