The switching power supply inside the monitor is operating in the human hearing frequency range and not very efficiently. This is caused by weak capacitors or resistors that have dried up or are getting hot. The longer you leave it, the worse it gets until it goes off completely. I have the same problem with one of my monitors but it's all high voltage or very high voltage inside so not a good idea to fiddle inside while it's on!
Try replacing capacitors and measuring the resistance of resistors in the power supply section. Resistors may read lower than expected, but should not be higher (fault?)