Does a Shinigami still die if they extend a human life without knowing it?
If the god of death is using the death note was killing at random, it would not result in the god of death dying. See rule XVII:


  If the god of death decides to use the Death Note to kill the assassin of an individual he favors, the individual's life will be extended, but the god of death will die.




The killing will only result in the god of death dying if they kill someone who is going to kill an individual the god of death favors.