that an incorrectly inserted RAM module would cause a burning smell. It would simply not be recognised when the machine performed its POST, and you would get a beep from the BIOS.
It sounds to me as if something on the motherboard may have shorted. You could try opening the case and checking every connection, then carefully inspect the motherboard for an signs of tracking (small black or brown trace marks). Do the fans run when you try to start the machine?