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I can't tell you why the might work but it is how I resurrected a 5,1 Mac Pro after pulling my hair out trying everything.

1. Remove all of add-on cards including the GPU and keyboard. The computer should boot without a GPU.
2. Unplug the computer for a minute.
3. Plug back in and turn on.
4. From another computer go to screen sharing. This won't work if you never set up the Mac Pro for screen sharing. Assuming that you have set screen sharing, see if you can see the login screen on your remote computer with screen sharing. If not, stop as this won't help. If you see the login screen from the remote computer, plug in a keyboard and restart without logging in, and hold down Command, Option P, R keys to resent PRAM and continue holding until you hear the start up chime 5 times. Assuming you get that far, plug in the video card and connect to your monitor. If you get to a normal reboot reassemble and go forth and prosper.

Like I said, I don't know why, but these old Mac Pros suffer from all kinds of crazy and this has worked for my with 2 machines. I am beginning to think it would be a good idea to reset PRAM once a month.
 
Has more to do with pin connections than overheating. It would happen within seconds of booting on mine. Fans go full bore. Black screen.
It happens to my 3.1 too. It happens only few minutes after boot if the computer survive it stays forever even running bechmark, pro software, games.
 
Having tried everything, I moved on. I sent the 30" Cinema Display to BuyBackWorld last week. I was quoted $40, but they paid for the shipping (which was easy, since I had the original box). Upon inspection, BBW found that there was a slightly darkened corner about the size of a nickel at the lower left of the display. So my payment went down to $15. Okay. It's out of my house, and BuyBackWorld will find someone who can use it.

Now for the tower itself. I'm pulling the two SSDs that I installed, and I may pull the upgraded graphics card. I also have a Mac-flashed GeForce GTX 980 Ti, which I should probably try to sell on the interwebs. I'm planning on taking the rest to the recycle bin of my local BestBuy in another week or so.

At least my 2017 iMac 27" is running well. My new computer is a PC assembled by iBuyPower, with a 32" Samsung curved monitor, running Linux Mint and, <cough, cough> Windows 11. At least the Windows side runs VR apps on Steam well. The new hardware is Intel Core i9-12900KF, 32 GB DDR5 RAM, and NVIDIA GeForce RTX 3070 (8GB) graphics. The new system blows my old computer out of the water. But I still feel nostalgia for the old Mac Pro -- and probably always will, just like my NeXT Cube.
 
It happens to my 3.1 too. It happens only few minutes after boot if the computer survive it stays forever even running bechmark, pro software, games.
I thought it was fixed but it seems to be more temperature dependent on my machine. Specifically in the morning when it's cold it will boot and then go dark, fans full bore. If I put a small heater on it or let it run like that it will boot normally once it warms up a little. I have heard the mosfets on the mobo are sensitive to temp when failing. Also I heard that booting it with fans disconnected then shutting it down and reconnecting them resets the temp control. I tried this but it didn't help. I am afraid the motherboard is damaged in some way.
 
I thought it was fixed but it seems to be more temperature dependent on my machine. Specifically in the morning when it's cold it will boot and then go dark, fans full bore. If I put a small heater on it or let it run like that it will boot normally once it warms up a little. I have heard the mosfets on the mobo are sensitive to temp when failing. Also I heard that booting it with fans disconnected then shutting it down and reconnecting them resets the temp control. I tried this but it didn't help. I am afraid the motherboard is damaged in some way.
OK damn... have you ever tried to change PSU?
 
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