Should Christians try to evangelize atheists?

As Christians who know the love of God and have the assurance of
eternity in heaven, it's hard to understand why anyone would want to
be an atheist. But when we realize the sin nature and its strong
influence on the mind and the heart, we begin to understand where the
atheist is coming from. Biblically speaking, there is no such thing as
an atheist. Psalm 19:1-2 tells us that the heavens declare the glory
of God. We see His creative power in all that He has made. Romans
1:19-20 follows up on this idea, telling us that what may be known
about God has been made plain to us through the creation, and anyone
who denies this is "suppressing the truth in unrighteousness" (v. 18).
Psalm 14:1 and 53:1 declare that those who deny the existence of God
are fools. So the atheist is either lying or he is a fool or both. So,
what is it that causes someone to deny God?

The main goal of those under the influence of the sin nature is to
make himself a god, to have complete control over his life, or so he
thinks. Then religion comes along with obligations, judgments, and
restrictions, while atheists presume to define their own meaning and
morality. They do not want to submit to God because their hearts are
at "enmity against God," and they have no desire to be subject to His
Law. In fact they are incapable of doing so because their sin has
blinded them to truth (Romans 8:6-7). This is why atheists spend most
of their time complaining and arguing not about the scriptural proof
texts, but about the "dos and don'ts." Their natural rebelliousness
detests the commandments of God. They simply hate the idea that
anything—or any One—should have control over them. What they do not
realize is that Satan himself is controlling them, blinding them, and
preparing their souls for hell.

In terms of evangelizing atheists, we should not hold back the gospel
from someone just because he or she claims to be an atheist. Do not
forget that an atheist is just as lost as a Muslim, Hindu, or
Buddhist. God certainly wants us to spread the gospel (Matthew 28:19)
and to defend the truths of His word (Romans 1:16). On the other hand,
we are not obligated to waste our time trying to convince the
unwilling. In fact, we are warned not to expend excessive effort on
those who are clearly disinterested in any honest discussions (Matthew
7:6). Jesus told the apostles to go and preach the Word, but He did
not expect them to stay anywhere until every last person had been
converted (Matthew 10:14).

Perhaps the best tactic is to give each person the benefit of the
doubt, at least at first. Every question, honestly and truthfully
answered, gives that person a chance to hear the gospel. But if that
person is just arguing, being hostile, or otherwise not listening,
it's probably time to go somewhere else. Some people are totally and
absolutely hardened to the gospel (Proverbs 29:1). They may be
rational or irrational, but there are scriptural reasons to believe
that some people are willingly immune to the influence of the Holy
Spirit (Genesis 6:3). When we have made a good-faith effort to talk to
someone, and he or she is unreachable, then we are commanded to "shake
the dust off" of our shoes (Luke 9:5) and spend our time talking to
those who are more spiritually open. As in all things, the wisdom of
God is crucial. God has promised that wisdom to us if we ask (James
1:5), and we should pray for it and trust God's prompting to know how
and when to end the dialog with a hostile atheist.