Try to do the following. When your GPU is installed, try and set your Power Saving settings so that the power button can sleep your computer and so the computer will not go to sleep after a certain amount of time. Next, troubleshoot the computer by removing the GPU, booting, and allowing it to stay on and running for an extended period of time. After a few hours or whenever you think is good, see if it will remain in sleep mode without completely shutting down. Since we have no video, thats where pressing the power button to sleep the computer comes in handy. If it works fine without the GPU, I'd say its time to put an HD 5770/HD 5870 in there, I think they're compatible with the 2006 model Mac Pro. If not a new PSU is super easy to install.
Let us know the results!