First thing to try is loosening the Screws that hold the Motherboard to the Case. They will only need a full turn. Reconnect the Power and turn it on. If it powers up shut it down and retighten the screws. They only need to be firm. Also check for loose screws under the motherboard. Check the power outlet that you are connected to by plugging in something else and turning it on.
Faulty Fans, Dust and Grime build up on the Heatsink restricting air flow. I would give the inside of the case a blowout with compressed air. When blowing air through the fans make sure that you physically stop them from spinning, as they may generate power and blow something up. Then remove the Heatsink, giving it a thorough clean and reseat the CPU applying new CPU grease. If the Fans spin freely when you give them a spin they are probably OK. If there is resistance replace them.
Clean the golden edge of each memory stick with a soft rubber, remembering not to touch the golden edge of the memory stick. Check with one stick at a time, remembering to disconnect the power from the PC.
Also check Capacitors around the CPU for swelling or bulging.
Note: A damaged or insufficiently charged internal battery can corrupt CMOS or BIOS settings.
If it doesn't startup:
Carefully remove everything from the Motherboard. Remove the Motherboard and put it on an anti-static mat on a bench. Remove the Power Supply. Remove the Heatsink and give it a good clean. Remove the CPU and reseat it. Reinstall the Heatsink remembering to use CPU grease. Clean the golden edge of each memory stick with a soft rubber, remembering not to touch the golden edges. Check with one stick at a time, remembering to disconnect the power from the PC between checks. Connect a hard drive. If you have access to a PCI Video Card use it.
Reinsert the 20-24 pin power connector to the Motherboard. Connect a Keyboard and Mouse. Connect the Monitor. Connect the Power to the Power Supply. You can use a flat blade Screwdriver to short the Power pins on the Motherboard if you haven't got a spare switch. Turn it on and see what happens. If it stars up add one Device at a time to test.
Also check the power outlet that you are connected to by plugging in something else and turning it on.