Just being really sad won't cause you to die.
But being really sad can cause you to lose your will to live which can then cause you to take your own life by suicide.
Or you may be so sad that you decide to never eat again and starve yourself to death because you need food to stay alive.
I've heard of a few people who starved themselves to death after a loved one died such as a spouse as the person just was so sad that they gave up on life and quit eating.
And the person slowly died as a result of being so sad that they quit eating.
Even some animals who become attached to their owners may quit eating when their owner dies.