Nowhere is the exact time or incident that produced the death of hope in the protagonist precisely referenced. However it is absolutely clear that there are a number of key, foundational moments which produce this death of hope in the protagonist. Of course, everyone reacted differently to the horrors of the concentration camp and people would have let their hope die at different times. However, one of these moments for Elie is definitely when he and the rest of the camp is forced to witness the hanging of three prisoners, one of them a young boy:
And so he remained for more than half an hour, lingering between life and death, writhing before our eyes. And we were forced to look at him at close range. He was still alive when I passed him. His tongue was still red, his eyes not yet extinguished.
Behind me, I heard the same man asking:
"For God's sake, where is God?"
And from within me, I heard the answer:
"Where He is? This is where--hanging here from this gallows..."
That night, the soup tasted of corpses.
This moment can be identified as the moment when Elie lost his faith forever in a God who could allow such atrocities to be committed, and implicitly we can identify the death of his hope. Of course, this is one of many possible moments that you could identify so you would do well to go through the text by yourself and pick out other similar moments that result in the death of hope in Elie.