Replace CPU with one that has a lower core voltage

What makes you think its bad? Chips come out of the same type and require slightly different Vcores.
If it isnt:
Overheating (Volts too high, use speendfan or coretemp to test)
Crashing a lot (Volts too low, system unstable)

Then i really would not worry about it.

If you do put the 2nd CPU in, just be sure to set the Vcore in the BIOS.

hmm i might have misread. you have a problem and you have diagnosed the P4 2.6 as being at fault?

Yes, the fault is either the motherboard or the processor (P4 2.6 SL6WH). I don't have another board, just another processor. I've swapped out the Power Supply and RAM, removed all PCI cards, cleared the CMOS, reseated the original processor, etc.

With the original processor the fans come on but I get no beeps, even if I have no RAM installed.

The question is, will the P4 2.66 (SL6PE / 1.5v) be fried if I put it in that board?

noperz, as long as you clear CMOS and check the Vcore in BIOS.