PC Powered Down and Won't Turn On. CPU Fan Was Very Dusty. Tried New PSU With No Luck.

Go back to the heatsink on the CPU. I am glad you found the heavy dust on the fan and cleaned it. But that tells me your likely have also a heavy dust accumulation in between the fins of the heatsink under the fan. On most CPU cooler systems if you are careful you can remove the few screws or spring clips that fasten the fan itself to the heatsink, without ever removing the heatsink from the CPU chip. Do that and see clearly what is in the gaps between the heatsink fins. I have seen some where the gaps are completely filled up with dust!. The gaps should all the clean so that air can flow through them (from the fan) and take the heat away. If your heatsink's gaps are plugged, clean them GENTLY, then re-mount the fan.

"Tripped" in the BIOS..." well, maybe. I don't think it works that way, but could be. You could check a couple other things, though.

Your post says, "I tried to power it back on, the case power light lit up momentarily, the CPU fan turned slightly, but no boot" right after the first shut-down. AFTER you had cleaned the fan, did it do exactly the same? The post makes me think of two things. First, assuming that the shut-down was because of CPU overheating, if you tried to re-start right away it would still be too hot and would not allow any re-start. But after an hour or so to cool down it should have started, even if it overheated again and shut down again. Secondly, does your fan still work? (How to check later)? If your fan does not turn and send a speed signal to the CPU_FAN header, the machine will not start up because it knows the CPU cooling is not working. This can happen two ways. One is simply that the connection of the fan to its CPU_FAN header is poor and the fan's speed signal is not getting to the header. For this you can just gently unplug and re-connect that fan several times. This action usually will "scrub" the contacts and clean off any oxides and dirt so the signal can get through. The more obvious one, though, is that the fan itself really has failed and cannot turn. Now, with a "normal" machine that is operating properly, you can just turn it on and watch whether the fan starts up and runs. You don't quite have that situation. But you can test another way. The fan's power supply is a max of 12 VDC. If you look at its connector, it probably has 4 holes in it in a line. On the side of it there are two ridges that line up over the plastic finger sticking up from the mobo at the CPU_FAN header. Those ridges are near the holes of Pin #1 and Pin #3; Pin #4 is outside the ridges. Now, Pin #1 is the Ground or (-) side of the power connection, and Pin #2 is the +12 VDC connection. If you get the fan off the heatsink, you can use small wires to connect those two holes of the fan connector to a car battery which supplies 12 VDC. Do the polarity right: battery larger post is the +12 VDC to connect to Pin #2, smaller post to Pin #1. The fan should start up right away. When you disconnect, the fan should slow down gently before stopping, not just quit real fast. If you fan can do that with a 12 VDC supply, then it is OK.

To clean the heatsink (assuming it is clogged) I would recommend NOT removing it from the CPU if you can avoid it. Removing can be difficult sometimes because the thermal paste can get hard. Even if you get it off easily, then you would have to clean off the old paste and apply some new paste before re-assembling. But most importantly, getting the entire heatsink and fan assembly off the CPU chip still does not give you easy access to the heatsink fins because the fan is still in the way. But if you can't get the fan off, maybe you would need to do it this way.

By the way, yes, a clogged heatsink can certainly cause the CPU to overheat and shut down. BUT if that is the only problem, normally you would be able to re-start after an hour or two of cooling off, even if it did not keep running very long. Your post suggests you can't even do that, which is why I suggested checking whether the fan is working at all, and whether the speed signal is being connected into the CPU_FAN header.

IF you think you need to reset the BIOS, post back here. I can give you the complete sequence to do this properly.