My outboard motor will not move up. How can I fix this?.

It seems like you may be experiencing a short in the neutral switch of your Honda outboard motor. This issue could be causing the engine to shut down when you push the throttle past trolling speed. If you're able to manually run the engine without the shifter and it runs full tilt, this could further support the possibility of a short in the neutral switch. You may want to have the switch inspected and repaired or replaced by a qualified mechanic to solve the issue.

no car make or model mentioned so getting an answer is hit and miss
however what you describe indicates a fan drive problem so if it is electric fan cooled-- check the fans for full blade speed ( weak motor operation) or if viscous fan hub ( fan clutch) replace the viscous hub
it is all about moving air through the radiator when the vehicle speed is slow and the engine rpms are high
if the fan is not doing the job of pulling air through the radiator , the car will overheat
at highway speeds the ram action of the air flow virtually makes a fan useless as there is more than enough air passing through the radiator to cool the water
Im guessing here but I suspect that you have a viscous fan hub

If the engine is revving up but not increasing drive, my thoughts are it is probably a prop drive problem.

check fuel filter, an for vacuum leaks at carb

There are videos all over YouTube that cover this subject quite nicely.
winterizing an outboard motor YouTube

try priming the fuel bulb when it boggs down and if it speeds up then replace fuel pump diaphrams etc and clean carbs and balance to start a fresh ! fuel tank clean ?

First, the over heating alarm is for YOU to shut the motor down...
Replace the water pump assembly..(have a different mechanic do it!!!!!!!!!!!!!!)... they are NOT all created equal!!

outboard motors some times are designed to not run at speed while tilted as the water pump inlet is above the water line and the engine will over heat and the pump impellor will be destroyed.. With an engine tilted the load from the propeller tends to lift the nose of the boat high and makes it unstable on the water . The best situation is to have the propeller tend to drive the nose down thereby promoting a planeing action that improves economy and speed Once the boat is on the plane you can cut back on the throttle and still maintain the same speed . Moving with the leg tilted should only be used to navigate shallow water at very slow speeds

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