I think the only way to remove that at all is to reshoot and make sure it's not there when you hit record.
Yes, that happens a lot. Try changing the refresh rate of the monitor.
So, just out of curiosity, why are you filming a monitor instead of using software to capture the monitor output?
The only option I have for capturing the monitor's output is something like Camtasia which can't run during a startup sequence, such as you see here. I'll experiment with the refresh rate to see if I can get any better results.
If the refresh rate doesn't help, try changing the frame rate of the camera to as high as it goes.
Also, if you move the camera closer, or farther away, using a zoom lens to compensate, perhaps the result will be different.
A last resort might be to apply a slight blur to hide the moire instead of eliminating it.
That moire is more likely caused by an interaction between the screen material and the sensor of the camera. My recommendation would be to try moving the camera closer or farther from the screen.
Baring that, try a different camera that doesn't produce so much moire.