Computer shuts off after switching to a gtx 580

Looks like your PSU is tripping the OCP.

Are you only increasing voltage? Or also increasing clock frequency too? Tell us exactly what you are doing. Exactly how much voltage and exactly what clock frequency and all other useful details.

Maybe the mosfets are overheating. Just a guess.
With my reference GTX 580, my overclock is limited by voltage. 3D Mark 11 will freeze if I attempt more than 1.113V

1.113V will also freeze if my 220mm side fan is less than 100% speed.
Perhaps you could remove the water block? and check the thermal pads which I assume should be covering the mosfets? Most GTX 580's do not have a mosfet temperature sensor, but you could check in gpu-z and see if your GTX 580 has one.

You could try blowing air onto and across the backside of your GTX 580 . Open the slot above your GTX 580 to allow air to exit. See my avatar as example.

Or maybe your cpu or gpu overclock is unstable. Heaven is good for testing your power supply and testing temperatures, but does not guarantee a stable overclock in all games.

It's hard for me to give you a definite answer because I dont know what graphics settings you use in Heaven or BF3 or in any another game. Still dont know eactly how much voltage you are use and when.

There are two things which will cause mosfets (VRM's) to become hotter: more voltage and/or more current.
stock clock + more voltage = hotter mosfets
stock voltage + more current = hotter mosfets
more voltage + more current = much hotter mosfets

High graphic settings (dx10/dx11, high antialiasing, high resolution, ...) should use more current than low graphic settings.

Do a OCN or google search for "hydro copper 2 thermal pad". I see something about "Don't forget to peel off the protective film from the new thermal pads". Or maybe your hydro copper 2 is not mounted tightly enough. Be careful not to over tighten it.

Here's something to try:
Play planetside again at 925MHz but turn down the graphics settings. Maybe low or no AA. Or turn off some other graphic settings.
Assuming the problem is caused by overheating mosfets, then with reduced graphic settings, planet side 2 should run longer without mosfet overheating.

Shogun 2 in DX11 causes my GTX 580 to overheat and freeze at 930Mhz on a warm day. Fortunately for me, I don't like Shogun 2.
The PC version of Alan Wake uses high resolution textures and in 3D will freeze if my side fan is set to low speed.
Last time I ran Heaven, I noticied higher gpu temperature in DX10 than in DX11. I know that sounds strange but that's what I saw.

Another thing to try:
turn up the radiator fan speed. That should help cool the mosfets a little more.

I don't know if you need new thermal pads. Maybe make some pics and show us.
Maybe someone on the watercooling forum can answer that question better than me.
There are videos of the hydro copper 2 on youtube which might help you to know if the thermal pads are installed properly.