David Grier, chair of the physics department at New York University, dials up an answer to this mystery.
This sounds like a case of electromagnetic interference (or EMI), which is what happens when radio waves emitted by one device cause undesirable behavior in another.
Virtually every piece of electrically powered equipment acts as a radio transmitter, whether it is supposed to or not. That's because the rapidly changing electric currents running through these devices naturally radiate electromagnetic waves. This is an inevitable by-product of using electricity to do useful things, and it is analogous to the clanking and clattering sounds that mechanical devices make as they work. Computers are particularly "noisy" because they rely on rapidly changing currents to act as clock signals that coordinate their calculations.
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Just as changing electric currents radiate radio waves, radio waves induce electric currents in conducting materials. This is how radio receivers detect the signals transmitted by radio stations. The same effect is used to heat pots and pans on inductive cooktops. Inductive coupling can also have undesirable consequences, however.
One explanation for the phenomenon you describe is that your computer unintentionally emits radio waves in the range of frequencies reserved for cell phone communications, typically around 800 megahertz (MHz). If the signal coming from your computer is strong enough, your phone could mistake it for a cell phone transmission. Computer noise, however, does not contain the sort of information that your phone's onboard computer is programmed to expect. Thus, it responds to the resulting cascade of communications failures by creating a series of audible alerts.
Another explanation involves a deeper connection between your two devices. In addition to its other components, a cell phone has an audio amplifier that drives its speaker, and the radio waves emitted by the computer may induce currents in the wiring of the amplifier itself. The resulting audio output then would reflect what your computer is doing at that moment but would sound to a person like random squeaks and squawks.
There is no way to stop electrical devices from generating radio waves. The only way to prevent EMI is to keep spurious radio waves under wraps. Most electronic devices are housed in cases designed to trap these electromagnetic waves; they are made of metal or have a coating that conducts. Holes in the cases and thin spots in the coating allow some radio waves to leak out. Usually the leakage is too small to have any effect except right near the source, where it is most intense. And that is why your cell phone only acts up when it is right next to your computer.
More on EMI:
The basis for the first PC peripheral: In 1975 Steve Dompier programmed his MITS Altair 8800 personal computer to play the Beatles's "Fool on the Hill." The trick was that the output device was a nearby AM radio. Dompier had programmed his computer so that the EMI created by the primitive, first-generation PC produced recognizable tones from the radio. The encore was a performance of "Daisy," which was also the first song ever played on a computer (at Bell Laboratories in 1957) and the swan song for HAL in the movie 2001: A Space Odyssey.
May jam electronics: The currents induced by particularly intense radio waves can be large enough to melt the wires in electrical equipment. This effect has been discussed, at least since World War II, as a way to disable electronic equipment.
Tips a computer's hand: The radio waves produced by computers are far from random. Nefarious folks have figured out how to detect and interpret a computer's EMI to learn what the computer is doing. This includes being able to "read" the sequence of keys being pressed on the keyboard and to "see" what's on the screen. Scary!
