Yamaha 40 outboard stuck at full throttle won't idle

Are the controls for the 'fast idle' loose? Also--run some seafoam through the fuel system on a regular basis--one per month.

Have your spark plugs changed AND carbs cleaned and adjusted.
Usually that can also be caused by a seized advance system ( plate under the flywheel, commanded to turn with the throttle linkage... )

have you left the choke open ? this mixes more fuel to the engine upon starting but will stall it if left open too long . check your fuel filter too it may be full of debris.

this sounds like the high idle circuit of carberetor has debri in the needle. I would consider cleaning carb and try again. If still continues then this would make me look at fuel pump. These 2 things are where you need to start.

You need to pull the cover and track where the trottle cable goes to on the motor. If you have someone open and close the throttle while you're watching you will find the linkage that it controls. There should be some kind of an idle ajustment screw there that you can readjust to increase your idle.

Reving in neutral does not mean the same thing as under load. If you need to pull 4500rpm under load and for an instant the load was gone, you would probably overrev to 6500. With the cover off and motor not running move the throttle to full forward. Look at your carbs and see if the throttle plates are wide open. If they are not all wide open take it back to the dealer. The throttle adjustment is off. You need the full range of travel from idle to topend.

Sound like the fuel pump is not keeping up with the engine's demand. Check it out. Good luck. Hope this helps.