This is the point where I'd get my scope out and check for clock signals and see if the reset line is behaving itself. Some of these old boards required the "Power Good" signal on the power supply connector; others didn't--so you may want to check that your PSU is supplying it.
Does the PSU fan keep running when you turn it on? If not, you've likely got a short somewhere on the mobo.
Try swapping the crystal for a known good one.
After that, I'd hang my scope on the address lines on the expansion slots and see if the machine is flatlining.
If not, check the position of the BIOS EPROMS--make sure that they're not swapped; make sure that the mobo EPROM jumper JP1 matches what you've got. If you've got an EPROM programmer, take a look at the EPROMs and make sure they're okay.
BIOSes don't usually require any RAM to sqeak out an error beep, so you might remove all of the RAM chips to make sure that they're not jamming a signal.
Check the expansion connectors carefully to make sure that the contacts look okay--I've seen mobos where contacts have been shorted against one another by some gorilla jamming something into a slot the wrong way.
Turn the board upside down and tap on it to see if anything falls out.
This is one of the reasons that I hate playing with the newer 286 and 386 mobos--if the support chipset is toast, there's not much you can do. I like the ones where all of the mobo is pretty much all known LSTTL commodity parts.