First thing to attempt would be reseating memory, then a memory swap for known good ram. Also try one memory stick in either slot, and no memory to see if you get a no memory error (some kind of beeps usually). Also try re-seating the cpu, and disconnecting all drives and cards (including graphics card) and also try a different PSU.
I guess you've tried all that... So, if it still doesn't work...
It's most likely that soldered joints on the board have become weak or fractured, especially if has been used for long periods of time. I see this with shop tills, which are on 24/7, when I have to shut them down to do work they sometimes won't reboot, especially if they've been crammed in with crap under the counter and getting hot due to lack of ventilation.
So it may be worth attempting to reflow the board, especially any parts which get hot, including power components. I have successfully used this method on 2 PS3's and my laptop, so it's definitely worth a shot. If you can get access to a reflow oven, that would be ideal. I use a heatgun, which isn't a very good method. If you do it this way you MUST practice on some scrap boards first. You want to melt the solder, but not scorch anything or have parts falling off. First remove the battery. Then flood all the areas you want to re-flow with no-clean flux, shield any plastic parts such as connectors and the bios battery holder using some foil (ideally something more substantial), ensuring it can't blow off. Warm the board first before concentrating on specific areas. Ideally you want about 350 deg C from your heat gun - mine has 2 settings high and low, so I use low. It's still a bit too hot.