In Night, Chapter 8, Does Eliezer feel guilt as his father dies? Why?

Eliezer feels more of a relief when his father dies.  By this point in the work, Eliezer's struggle to survive has taken such an encompassing hold on him that the death of his father is seen in survivalist terms.  Eliezer believes that it will be easier for him to live in the camp without having to worry or tend to his father.  Additionally, given the fact that his father was in such dire physical condition, Eliezer will no longer have to expend energy and care in tending to him.  If there is guilt, it is in the feelings of not having any guilt, especially given the promise he made earlier that he was going to do his best to maintain the bonds between he and his father and not break them.

It should not be read, I don't think, as Eliezer has become unfeeling or selfish.  A concept brought out throughout Night is that the horror of the Holocaust was in its betrayal between people, and that in certain conditions, bonds of loyalty are quickly broken between individuals.  The traditional belief is that people always rise to their own sense of best during harsh conditions.  Yet, Wiesel is astute to point out that the real terror of the Holocaust was that sometimes the best of nature is not revealed, but actually the worst.  Recall the previous scenes how people stepped and fought over each other for a piece of bread.  Seeing this is where the real horror of the Holocaust lay.  In recognizing this, Wiesel is making an earnest plea that such a condition, in any form, must be defeated and opposed at any and all costs.  In another portion of the book and later on in his life,  Wiesel recalls an incident from his own life where he saw a woman throwing coins at children who were fighting one another to obtain them.  When he asked her why she does it, she responds, "I like to give to charity."  In presenting this image, the reader is confronted with the understanding that human indifference and savagery might have been a condition of life during the Holocaust.  Yet, the only salvation from such a period has to be the changing of moral behavior and lessons from it.  When this woman throws coins to see children fight and responds with her supposed "love of charity," Wiesel might be saying that this might be on the same level of repugnance as what was experienced during the death camps.