I have a 2010 Mercury Optimax. I did not use it for about 3 months so installed a new water pump.Took it to the lake to run it and got good water pressure while running and no beeps, however, while sitting in neutral and bring it up to about 1600rpm a solid beep goes off until I bring it back to idle.Is that normal or a warning for overheat? Ran it again about 5000rpm and no warning beep.

You`re going to need to hook it up to a scanner to tell but ones I have seen this in the past can be the compressor over temp, the fuel pressure regulator diaphragm or throttle position sensor.

I read all these stories on outboard problems and I am not really sure it has been progression in the marine world.

You are activating the neutral overspeed warning.....OPTI

If it has a silver lower unit with the holes in the front you may have picked up some trash and it is lock up in the thermostat.It is a TQ master 2 Worst lower unit ever. IMO

When I ran the silver L/U I had to pull the thermostats 12-18 times a year with trash being stuck in them. Water pressure will show good but motor only reads one side.

If you bump stumps this most likely is the problem. When the thermostat is stuck open it will sound a alarm at idle or slow idle but not running across the lake very often.

I run a standard lower unit now days after learning all that.

I would bet you have trash in the thermostats. IMO