How old? More than 5 years? More than 10? Best to list the specs of the system, for a better idea of any problem causes.
First check the status of the BIOS battery. With the AC power lead pulled, check the battery voltage in-circuit. A digital meter is preferred. The top of the battery should be marked with a "+" sign, and any bare chassis point such as metal shields or a black lead terminal is negative. Should show a +3 volt level, if lower than 2.8 volts it is at its end of life. Typical life is about 3~4 years, depending on use. It is only a backup for when no AC power enters the PSU and the PSU provides a +5 volt stand-by source for BIOS/CMOS and USB circuits. The real-time clock is part of this circuit also, and if not functioning, the motherboard may not show any activity!
Replace the battery, reset the BIOS pages and it should be ready for use. First signs of a BIOS battery dying are the date and time wandering off.
Could also be a case of capacitors going bad. Look at all the capacitors on the motherboard.. if any show domed tops, lean excessively, or exhibit signs of leaking, they are bad. More info at www.badcaps.net/
As to the monitor being damaged by the system video, very unlikely. Testing the monitor on another known-good system will reveal its status. With flat-screen monitors, there are thin fluorescent lamps behind the bezel, which give the screen its illumination. They have a finite life, like any other fluorescent lamp. Those lamps are powered by a small inverter board, and it too can go bad. Try connecting the monitor to a known-active video source, with motion graphics. Shine a flashlight into the screen, and note if any activity. If so, then bad lamps or inverter. But the monitor has a power LED indicator that should be green if an input signal is good, amber if no signal. What status do you have?
F@H.. to solve mankind's maladies.. in our lifetimes!