They cannot die from Get to Work illnesses. However, that illness that causes sims to giggle may lead to death by laughter if sim has too many happy and playful moodlets. (It happens rarely and can be avoided by making your sim sleep or taking medicine)
Jungle Adventure allows your sim to be poisoned by various ways. This leads to death if antidote isn't found within 4 days. Poisoning may go away on its own after 12 hours.
Finally, My First Pet stuff pack adds Rapid Rodent Disease. This illness can come when sim gets bitten by a rodent in an unclean cage. If sim doesn't get cured within three days, they'll die. On the third day, the disease can spread to other sims that get near them.
