So far I have looked at the explicit text to determine whether 'we' are to do evangelism. The evidence to me is overwhelming; essential to Christianity is the imperative to share the gospel with the lost. This is seen in the sweep of the OT narrative, the ministry and teaching of Christ, the ministry of the first Christians through Acts and Epistles.
There are also theological strands that come together to reinforce this point:
1. Fall, sin, death and eternal destiny: it is clear through the NT that Christ, Paul and the other apostles worked from the premise that the Fall of humanity in Gen 3 led to all humanity being separated from God, under sin, facing physical death, and eternal destruction. There is only one solution in the bible given, Jesus Christ. His death and resurrection is God's solution to the problem of the Fall. He was without sin and died resolving the problem. He reconciles us to God. He redeems us. Through his death we are declared righteous if we believe. He is the saviour of the world. All humanity outside of Christ is lost, facing not only physical death, but separated from a pure perfect God in their sinfulness. The exact nature of eternal destruction is not clear, it could be a form of annihilation or some eternal existence apart from God. Either way, those who do not know God are lost. This reality screams out to all of us who have found salvation in Christ that we must do EVERYTHING we can in word, deed and attitude to see people drawn to Christ. While there are people who are separated from God we MUST be concerned for their salvation! How can we not be?
2. The selfless essence of the faith: Christianity is about selfless living for the needs of others. This is demonstrated in a God who did not need to create but chose to because he wanted volitional creatures to love and walk in relationship with. This selfless God is seen through OT as time after time he intervened for his people Israel (Judges, Exodus, restoration from exile). This despite their terrible failures. The ultimate demonstration of this is Christ who left his pre-existence glory to become human, experience the totality of human pain and suffering, die on a cross refusing to use coercive force in his own defence, and then rose from the dead. He remains human, God in a human body for eternity to save us! Selflessness involves deeply caring for others at their deepest point of need. The deepest point of need is one's eternal destiny. Selfless concern for others will involve us utterly sacrificing ourselves for the needs of others in this world leading us to care for the poor and needy. Even more it will lead us to care for the needs of others who are lost eternally if they do not hear and experience the message of Christ.
So we are to be evangelistically inclined. We are to look on those who we mingle with from the world and we are to meditate upon them, their destiny, their need, the horror of what they face. We are to focus our hearts on them, pray for them, show them love, share the faith, do good works that demonstrate that we care. We may face death and ridicule. We may be laughed at. We may be as the first Christians were, thrown into prison, beaten, burnt alive, cut in half, crucified, murdered... but this cannot deter us as it did not deter them. We must share the message with whole lives given over to God!