Whenever I play ACNL, I find my villagers are sick, but if they get too sick, can they die? I have tried testing this on an undesirable but forgot to play and when I came back they were gone from my town. Although, before they got sick they had been planning to leave so I think they may have just left.
The sickness will last from one to fourteen days, or until you give them medicine.
Your villagers will never die.