Question: "Did God die? If Jesus was God, and Jesus died on the cross, does that mean God died?"
http://www.gotquestions.org/did-God-die.html

Answer: Did God die when Jesus died on the cross? The answer depends on how we understand the meaning of the word die. To die does not mean to go out of existence. Death is separation. Physical death is when the soul-spirit separates from the physical body. So, in that sense, yes, God died, because Jesus was God in human form, and Jesus’ soul-spirit separated from His body (John 19:30). However, if by “death” we mean “a cessation of existence,” then, no, God did not die. For God to “die” in that sense would mean that He ceased to exist, and neither the Father nor the Son nor the Holy Spirit will ever cease to exist. The Son, the second Person of the Trinity, left the body He temporarily inhabited on Earth, but His divine nature did not die, nor could it.

Jesus is truly God and truly man. His physical body did die; His heart stopped beating, and “he gave up his spirit” (Matthew 27:50). Just as our physical bodies will someday die, so did His. But as Jesus died physically, He remained alive spiritually. Jesus made good on His promise to the believing thief on the cross—He and the thief both went to paradise, not physically, but in spirit (Luke 23:43).

So, Jesus died physically on the cross, as was plain for all to see (John 19:34). But there is also the matter of spiritual death. When Jesus was hanging on the cross, He experienced death on our behalf. Even though He is God, He still had to suffer the agony of a temporary separation from the Father due to the sin He bore. After three hours of supernatural darkness, Jesus cried, “My God, my God, why have you forsaken me?” (Mark 15:34). So, could it be said that Jesus also “died” spiritually? Again, it was only His human nature that was separated from God, not His divine nature. God did not “die.”

The question for all people to ask is “what will happen to my soul/spirit when it leaves my physical body?” This is the most crucial question in life. As we saw with Jesus, our spirits will leave our bodies and travel on to somewhere else. We will either follow Him to heaven to spend eternity with Him, or we will go to hell to spend eternity in “outer darkness” where there will be “weeping and gnashing of teeth” (Matthew 8:12). There is no other option. Jesus’ death on the cross paid the way for all who would ever believe in Him so that we can know for sure where our spirits will reside for eternity. His death provided us spiritual life, both here and in heaven. “And if the Spirit of him who raised Jesus from the dead is living in you, he who raised Christ from the dead will also give life to your mortal bodies through his Spirit, who lives in you” (Romans 8:11).

Recommended Resource: "I Don't Have Enough Faith to be an Atheist by Norm Geisler and Frank Turek"