Very high-pitched noise when computer does something intense?

It's called "coil whine". It is not harmful, just annoying. You can learn more about it on Wikipedia:

These coils, which may act as inductors or transformers, have a certain resonant frequency when coupled with the rest of the electric circuit, as well as a resonance at which it will tend to physically vibrate.

Basically you can not do much about it, some coils just have it, especially on graphics cards.

The two most likely culprits are coils and capacitors in the voltage converters in the various switching mode power supplies in the computer.

Switching power supplies use coils to convert power efficiently from input voltage levels to output voltage levels. The power supply applies a higher voltage (e.g., 12V, not really extremely high voltage) to the coil connected to a lower voltage load, such as the 1.2 volts or so used by the memory or CPU. The high voltage causes a current to build up in the coil. Then it disconnects the high voltage from the coil. The collapsing magnetic field in the coil opposes changes in current, so the coil tries to pump current to its load. A diode from ground on the input side of the coil conducts, so current continues flowing to the load after the high voltage input is disconnected. But the current actually flowing into the lower voltage output decays more slowly than it was building up when the higher input voltage was applied. Thus, you get more current at lower voltage. The output power is always slightly less than the input power, but the conversion can be pretty efficient.

The magnetic field in the coil acts on the current in the coil, producing an actual mechanical force on the wire in the coil, like the force produced in any electric motor. Since the magnetic fields and currents are changing, this causes changing forces which can vibrate the wire of the coil. Power converter whine is likely to be louder at heavier processing loads. Processors really do use more power when they are busy.

Some capacitors also sing. The voltage applied across the very thin insulator in the capacitor physically squeezes the insulator, compressing the capacitor. Ceramic capacitors use piezoelectric materials between conductors and actually store energy as mechanical strain. This effect is used to produce beeping sounds in some electronic devices, such as the beeper in a microwave oven that tells you that your popcorn is scorched into inedible charred fluff.

If your display uses cold cathode florescent backlight, there is a similar converter to convert to a very high voltage (e.g., 1200V) at a very low current. If this converter is the noise maker, changing screen brightness may affect the volume of the sound.

In one computer I had a few decades ago the sound card picked up electronic noise from the various components in the computer. It was only audible at high volume levels, but it was the limiting factor of the dynamic range of the sound card. Disk activity was a big factor in the electronic hash due to the large current spikes when the disk heads moved. If hash in the sound card is the noise source, reducing the output volume should affect the volume of the whine.

As some of the other answers suggest, this appears to be coil whine. I occasionally experience the same issue and was determined to find out what the cause was. I removed my Samsung SSD830 and attached it dangling from an eSATA cable outside the machine so I could place my ear next to it.

It was the hard disk.

I was a bit puzzled by this, seeing as there are no moving parts in an SSD. I wrote a CPU bound program to reproduce this behaviour and could see that the HD was idle during the run. This puzzled me further. However, it turned out that this behaviour manifested itself only when the laptop PSU was disconnected and the machine was running from battery only.

It may be that when CPU load increases, there is not enough current available to power all components without a voltage sag in another component, hence causing the behaviour you have experienced. Powerful GPUs and CPUs crave current like a flux capacitor, so it may be you need a higher-rated PSU to deliver enough current during load spikes.

Note This is all conjecture and just based on guesses, so don't go and buy another PSU without confirming the source yourself. This may however help you track down the source of the issue.