Pull out all the RAM and disconnect all devices besides the power supply and the power button.
(I've had several systems do this due to RAM that came loose)
Once this is done, try and boot it up. If you still don't even get warning beeps about no RAM, its either the power supply, the motherboard, or the button.
There is a simple power supply check you can get for about $20 at Comp USA to eliminate the power supply (or you can swap in another board, etc).
If you have a light on the MB though, the power supply is probably good. I've only ever known a power supply to work or not work. I've never really seen it somewhere inbetween.
A button on a case is just a simple switch. As a result it should be easy to simulate with two pieces of wire, and join the two wires together to trigger the "press".
However, I almost bet you $$ its your RAM, and taking it out you will hear beeps when turning the system on. You can then add a RAM chip at a time back and see if its a bad chip, or they were just slightly mis-seated.
I have a system now that every time I move the box I have to re-seat the RAM or it does exactly what you are talking about.