As per the title, if I'm starting a motorcycle or lawnmower or something similar with a carburetor and I know the mixture is too rich, does opening the throttle lean the mixture out? I've got a klr650 that I can't adjust the pilot screw on yet and it's starting without any throttle or choke when cold, but refusing to start when warmed up.
No (or maybe more accurately – not for long), that is why you have a choke. Opening the throttle does allow more air in, but should also be causing more fuel to come in as well to preserve the mixture. It sounds like you might have a stuck choke or perhaps your idle adjustments are off.
That said, when the engine is stopped, there is no airflow to pull in fuel, so you might get a momentary leaning of the mixture if you open the throttle when starting. Be prepared to close the throttle right away, as it won't be good for the engine to rev up so quickly before there is good oil delivery.