I just wanted to say that I move around computers that are on all the time and I haven't noticed any computers getting damaged by this ever. I tilt computers and lay them on their side without turning them off and never any damage, the same for standing them back upright from either of those positions.
I have half my PSU wires routed outside the case hanging off in the air and I can move those all around while it is on and nothing happens too.
Mind you I wouldn't advise other people to do what I do, but I never heard of anyone blowing a hard drive, PSU, motherboard, or anything else by laying a computer gently on its side when it is already on.
I have even done really stupid things like unplugging SATA or MOLEX power cables while the power cable is still in between the wall and the computer. I really try not to do that but once in a while if I slip up it never damages any of the parts for whatever computer I am working on. Again, I wouldn't advise anyone to ever forget to do that, though.
As long as you don't pull out cords while the computer is actually booted up you should be fine.
I highly doubt that the OP broke anything by moving the computer around.
I wouldn't advise anyone to turn a computer on with no heat sink and fan attached either, but if the OP turned the PC off within 10 sec (preferably within 5), then there is probably no damage to it. He should be able to get the oil from his fingers off with a cleaning solution and it should work just fine.
TBH, it doesn't sound like a motherboard issue to me. It sounds like a PSU issue.
Every single time I have ever seen a computer turn off and not turn back on it was because of a faulty PSU. The dozens of times I have seen computers turn off and it wouldn't go back on it was fixed right away with a new PSU.
I would suggest that the OP try a different PSU in the computer that doesn't work. One that is good enough quality to power the system, obviously, once he gets his finger oil off the CPU.