flat panel monitor is it? or an old crt?
modern flat panels usually have a standby mode which activates when no signal is coming from the pc ... this standby mode uses around 0.5w per hour.
your pc has power settings that you can set to turn off monitor after 1m-never.
a better option would be hibernate = the computer goes into an ultra low power mode, your monitor will not get a signal and default to standby.
Leaving monitors on can result in screen burn (crt), and in flat panels a similar pixel burn may happen (but pixel burn takes a very long time to occur usually and is random).
what happens is the pixel temps end up damaging the pixel crystal resulting in a 'burn' the colour of the pixel may end up fixed/dead/etc. but...