Is it the job of a Christian to prove to a non-believer that God exists?
It depends who you ask.
In II Timothy 2:15 [NKJV], believers are told to "be diligent to present yourself approved to God, a worker who does not need to be ashamed, rightly dividing the word of truth." In I Peter 3:15 [NKJV], believers are also told to "always be ready to give a defense to everyone who asks you a reason for the hope that is in you." Do these verses imply that a believer is to provide extensive evidence of God's existence, though? Or does the burden of proof lie with God Himself?
In Romans 1:20, Paul talks about evidence for God being found throughout creation. In Psalm 19:1, it states "The heavens declare the glory of God." Often, though, I've heard the statement that someone refuses to believe in God until they have sufficient evidence. I've also heard that many do not want to believe on the basis of the natural world alone.
Although these are understandable positions, what, then, is the threshold for intellectual belief? If one hundred people came up to them and offered their testimonies of what God had done in their lives, would there be one hundred explanations as to why it could not have been God?
To put it another way, if a person goes to the grocery store and picks up a case of soda pop, do they first have to visit the bottling plant and speak with all the representatives of the bottling company? Do they have to visit the factories where the aluminum and cardboard are produced that is used in the final packaging? Or is it important to go one step further and visit a bauxite mine, speak with all the employees of the mining company, and tour the forests which are cut down (and eventually turned into cardboard)?
What about all the equipment used in logging and all the machinery used in mining? What if the cardboard is made up recycled materials?
At some point, it comes down to faith. In Hebrews 11:6 [NKJV] it reads, "But without faith it is impossible to please Him, for he who comes to God must believe that He is, and that He is a rewarder of those who diligently seek Him." In the parable of the mustard seed Jesus talks about how even a tiny amount of faith can produce much in the hands of God.
To take it a step further, in John 14:21 [NKJV], Jesus states, "And he who loves Me will be loved by My Father, and I will love him and manifest Myself to him." In Jeremiah 29:13 [NKJV], God told the Israelites, "And you will seek Me and find Me, when you search for Me with all your heart." It is interesting to note here that God was not asking them to go halfway in their faith or to build their faith on a set of lengthy conditions. Either the Israelites were going to go "all in" and find God, or they weren't.
Perhaps underneath all of these debates it really comes down to a matter of trust.
Trust is needed in so many aspects of life in our world that without it, it would be impossible to do something as simple as going into a grocery store and picking up a case of soda pop. It would be impossible for countries to stay at peace and impossible for scientific progress to move forward. The difference between the case of soda pop and the Bible, however, is that God will provide continuing evidence of the truth of His Word and His existence long after the twelve pack is gone.