PRAM Battery?

Somewhere on your motherboard there will be a reset button—it may be marked "Reset", but may not be. The button will look like a small metal square, about 1/4" by 1/4" square, with a small—probably black—plastic circle sticking up from it. Press the plastic circle, the button, to reset.

Long shot, but try removing both DRAM DIMMs and power the machine on.

You should get a continuous blinking white LED. Now power the machine off, then refit the DRAM DIMMs and try again.

If that doesn't work, then I'd be looking at the Power Supply. Note there is a Power supply which provides 12v to a smaller DC-DC board just above it.

It's difficult to say what is really at fault, since anything in the iMac could be placing excessive load on the power supply, preventing it from powering up (though that is less likely than a failed main power supply).

Do you see any lights at all when it fails? IE, the white light at the front, bottom left?

Unfortunately, there is no reset button on the 17" 1.83Ghz Early 2006 iMac, nor are there any "Diagnostic LEDs".

Does sound like the main power supply, but that's just an educated guess.

The main power supply is Apple service part number 661-3780 which ironically, is the same main power supply in the iMac G5 iSight (the last G5 iMac).

All the 2006 iMacs (EXCEPT the 24") use the same main power supply.

The DC-DC board is different, and it's PN is 661-3878.

If you can get the machine open, and check the output of the main power supply, you should get 12 volts always across the black & grey cables.

WARNING. The main power supply is not enclosed once you get the LCD display out of the machine and there are lethal voltages present on the power supply. Even if it has been unplugged for a while, it's entirely possible that in it's "failed" condition that you could receive a powerful shock by touching circuit traces on this power supply.

I've seen people replace the main power supply with the 12v feed from an ATX power supply. This would require a bit of creative work and soldering to perform, but could at least rule out anything else.

As a final note, given that this iMac is Core Solo CPU, you might consider an upgrade. Obviously, compare prices of working Core 2 Duo machines with the cost of a used power supply bearing in mind that this could be a common failure.

One personal note, I have an iMac 20" Late 2006 Core2 Duo and I've had to disable the video acceleration driver kernel extension just to keep it going. Many of these 2006 machines suffer from failed GPU memory buffer which is NOT a separate card and requires a motherboard replacement which is far too expensive.