I am glad you asked and that iFixit provides a facility to reply.
Mac Pro - Mid-2010
If I let the machine sit unplugged for a week. It would then start, but would automatically shut down after several hours and never re-start... until say, another week had elapsed.
I am technical from a software perspective and I think I can fix anything... however, I failed personally and then took the time to find someone who would not just swap the logic/mother-board and call it fixed. Finding someone was not easy. I live in Canada, and this will sound harsh but all Apple certified places that I took the machine to were providing swap and replace services. I did not not want to do this for several reasons. They quoted the cost of a thousand for the logic board from Apple, then their labour.
I have two (2) identical machines. One was never used and the other is used regularly. The never used machine was the one that failed. Once, I decided I was going to fix it, I swapped all the components between the two machines and found that it was not the components. Then I took the logic board/ mother-board out and cleaned it with electro solvents, because sometimes the machine would work for say a day, then fail to start... for forever. I suspected that the problem was either capacitor or power. Cleaning did not resolve the problem.
I finally found a company that would do on-board repairs. They will charge $400 to replace the power chip.... and guarantee their work for three (3) months. I will get it back next week. I will post here to confirm that it is fixed.
Normally, I use high quality surge protectors but I am thinking that this machine may have been plugged directly into the house power system... we have electrical storms here, hence the power chip failure. I am guessing.
If you have questions, please ask.