Well, for one it is so bad for our health. The brain needs sleep to heal and repair itself. The body can also fight illnesses more effectively with sleep. Without sleep the body's immune system is compromised so it can make you ill. Your risk of diabetes, heart disease, and even cancer is greatly increased. It becomes extremely difficult to think and recall memories or even function. It can make you hot tempered, too.
While I don't think someone can necessarily die from no sleep - they are likely to die from hurting themselves doing something while sleep deprived like driving, operating heavy machinery, drowning in a pool/bath/shower, etc.
So they would fall asleep eventually whether they liked it or not as you have mentioned. It is an essential bodily function and the body will get it one way or another.