Even when the machine is "off" parts of the board are energized with +5V. (That's why you turn off the power supply itself before messing with the board.) I have seen that +5V leak onto motherboard USB ports. And I've seen the reverse--a powered hub leaked +5V back onto the motherboard--I had a power light on the motherboard with it's power supply unplugged. While I have never seen power leak in and back out I wouldn't be willing to say it's impossible.
One thing I would test: With the case open pick a small fan and carefully watch it, then push the power button. If the fan moves a bit but the machine doesn't come on the power supply isn't putting out enough power and you're getting an out-of-spec shutdown. I've seen this more than once with aging power supplies. Also, sometimes the machine will come on if you unplug as much as you possibly can from the power supply. Note that such a power supply will pass on every power supply tester I have ever seen--they only test at quite low power.
Disconnect EVERYTHING from the power supply except Motherboard & RAM,. This includes disk drives, USB devices, etc and see if it turns on. Leave the front panel connector attached too so you can use the power button. This will tell you if the computer can power on. Another option is to disconnect the Power Supply completely from the MB and peripherals (you can leave it in the case) and manually turn it on via temporarily shorting the pin from the green wire to any black wire. This will tell you if the power supply can at least turn on independently. – Damon – 2018-01-04T05:00:39.713
It could be that the PSU is not delivering enough power. The cheapest way to test the PSU without having a second setup to test the PSU, is to use a multimeter. – DrZoo – 2018-01-04T05:48:05.140
You can also purchase a PSU tester unit. Basic ones cost about $20. When you turn on the computer, does it beep at all? do any fans spin, even if just for a second? If you completely disconnect the power for 30 seconds and plug it back in, is there any change in behavior? – MaQleod – 2018-01-04T06:25:27.393