Hi,
I've been studying this and have come to the conclusion that the only way that the warning buzzer can activate without the engine running is that the Yellow w/Black stripe wire for the buzzer is unintentionally grounded.
This could be because the wire has rubbed/chafed through somewhere and is touching ground OR that the ignition control module is faulty and is providing ground when it shouldn't.
If you locate the Y/bl wire at the ignition control module (CDI) connector you can identify the pin on the module to test. Simply unplug the connector and ohm check that pin of the module to engine ground. The result should be infinite resistance. If there is continuity to ground then the module is defective.
If the module is good then you will need to find where the Y/bl wire is grounding.
If the module needs replacement to cure this problem it is available at boats.net for $147.21.
Item #2 in the link below.
https://www.boats.net/catalog/honda...-lrta-2007-and-later/ignition-coil-c-d-i-unit
However, if the engine starts and runs ok and you can't absorb the expense you could simply disconnect the buzzer. You can do that by locating the plug-in connection for the Y/bl in the wire holder at the top, rear of the engine. The individual wire connectors are all in that plastic slot holder located at the front of the empty plastic tray. It looks confusing but it's pretty simple to trace the wire and locate the plug. As long as you keep it neat when you put things back the wires don't necessarily have go to back exactly as they were.
If you do decide to run the outboard with the defective module, besides having no audible indication of trouble you need to also keep in mind that it could have other internal problems. Specifically, you really wouldn't know if the engine protection portion would provide a power derate and or shut down function in the event of an overheat or low oil pressure situation.
If you want to verify that the warning light circuits work that is possible but you'll need a 100 ohm resistor for the temperature sensor circuit and the engine would need to be run to supply power for the lights.
Anyway, I hope this helps you.
Good luck.