I've got an HP Pavilion that recently died on me.  When I turn it on, it produces a steady tone that's really annoying.  Nothing shows up on the monitor, no fans start spinning, no hard drives start spinning.  Basically, it just sits there emitting this tone.  Does anyone have experience with this or know what it might be?

Sounds like a psu problem.

The LED on the back of the PSU lights up fine when it's plugged in and as I said before, there's that tone whenever the computer itself is powered on.  I don't know much about hardware, but doesn't there have to be power for it to turn on the light and beep?

Yes, however a failing psu and a dead psu are two different things: one produces unpredictable problems that often lead people to improper conclusions, the other results in a functionless system.  It's unlikely for the BIOS to be at fault as it's warnings are simple morse code-like beeps rather than a neverending steady beep.  Unless there's something leaning on a key of the keyboard during boot up, I'd almost say positively that it's psu problem.

Okay, first you have to take everything about; Use the motherboard,memory,videocard,and the PSU. Then try Trial and Error , try different hardware (mobo,vc,memory,psu) if none of them are it, then it might be your video card. Thats probably the best thing to do. BTW, your supposed to take the stuff out so that, you can be sure that nothing is touching the steel.

Find out what's causing the sound, or use your warranty (if you've got one).