I have the same issue on my 2003 Avalon XLS. During the first 3 years, these lights have come on when I filled the car with gas, while the engine was running.
The dealership turned them off with no problem.
Later, (car is now 10 years old) the lights came on and stayed on. I did a diagnostics check at a toyota dealership who performed 99% of all the service on this vehicle. Im told (diagnostics report) its the charcoal canister, an emissions part under the hood. The dealer quoted $600+ for the repair. Im purchasing the part online for $200 and having it installed for $100. Maybe getting a diagnostics test will help.