[ 1. IBM 5150 motherboard of type 64KB-256KB.
2. IBM 5150 PSU of the type where the fan is DC powered.
3. The PSU's fan is not turning when the PSU's P8 is connected to the motherboard.
4. The PSU's fan turns when the PSU's P8 is disconnected from the motherboard. ]
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IBM 5150 (IBM PC) motherboard - type 64KB-256KB
This is the later version of the 5150 motherboard. It is marked "64KB-256KB CPU".
On this motherboard, there is only one component on the +12 volt line and only one component on the -12 volt line. Both components are tantalum capacitors. Click here to see a photo that shows the two capacitors.
The two subject capacitors are not critical to the operation of this motherboard, and this motherboard can operate without them. The reason that these two capacitors are not critical to operation, is because, on the 5150 motherboard (of type 64KB-256KB), they filter only the 12 volt lines going to the expansion slots. More information at here.
So what I am about to get you to do is remove the two capacitors, reattach plug P8 from the power supply to the motherboard, then see if the power supply's fan now turns. If the fan turns, one (or possibly both) of the removed capacitors is short-circuit.
Remove the two capacitors either by desoldering, or by using a pair of sidecutters to cut the capacitors off the board. If you have little soldering experience, then perhaps (if possible) use the technique shown at here, because on a 5150 motherboard, soldering the ground and/or power lines on the PCB is problematic, and your inexperience could end up damaging the PCB.
Now that the capacitors have been removed, reattach plug P8 from the power supply back into the motherboard, then turn on the power supply.
If the power supply's fan now turns (that being my expectation), one (or possibly both) of the removed capacitors is short-circuit.
If nothing else is wrong with your IBM 5150, your 5150 will now start.
CAPACITOR REPLACEMENT
As stated earlier, on this type of IBM 5150 motherboard, the two subject capacitors are not critical to the operation of the motherboard.
Should you wish to put in replacement capacitors (the recommended action), then click here for replacement details.