I have a guitar amplifier and I am wondering if I could connect any of its outputs to a PC microphone input on the motherboard through a 6.3 mm. to 3.5 mm. adapter. Or it will not work and theoretically burn down my motherboard?
To be honest, this is asking for trouble. I strongly recommend you purchase and use a DI box to isolate the amplifier from the computer. You are likely to experience hum and the DI will help to isolate. As @hobbes mentioned, under no circumstances connect the speaker outputs to the computer unless you need to buy a new computer.
The only connection that will have a chance of working will be the loop send, but it is not clear what level this is intended to work at, so I would definitely recommend using a DI box.
Loop Send should work. Try at first with the gain closed on the amp, and the microphone level on the PC set to a low level. Check the signal level, increase level until you get a good signal (peaks of -10 dB) without clipping.
Do not connect the PC to one of the Speakers outputs.