Calvinism is the term applied to a belief in a high view of the sovereignty of God, especially as it relates to salvation. Calvinists are convinced the Bible teaches that man is sinfully corrupt throughout his entire being and cannot make himself acceptable to God through any amount of effort of his own. Calvinists hold that in eternity past God chose out some among mankind for His own. In the course of time, God grants repentance and faith to His elect so that they might be awakened to their sinful state and need for grace. Those He saves will be preserved for eternity by the Lord and will persevere in following Him; i.e., if they truly belong to Him, they cannot and will not ever fall away because He keeps them secure.
The point which causes some to believe that evangelism isn’t important is that of "limited atonement." This point of Calvinism teaches that Christ died only for the elect. The theological argument offered is, if Christ in fact died for every single human in world history, then no one would go to hell since their sins are already paid for. Since we know Scripture teaches many spend an eternity separated from God, it must be that their sins were not covered in the atonement. Either that or there are people in hell for whom Christ died, a scripturally insupportable conclusion.
Some may say, "Christ paid for the sins of everyone, but it’s up to each person to decide for and accept Him." That’s the whole issue between Calvinism (monergism) and Arminianism (synergism). For if man casts the deciding vote, then how is God sovereign? Furthermore, if Christ’s sacrifice needs man’s acceptance of it to validate it, then it can’t be the all-sufficient sacrifice the Bible says it is. (See Romans 5, Ephesians 1:3-14). The Bible tells us that we love Him because He first loved us (1 John 4:19), not the other way around.
But Calvinism, and most anything else if out of balance, could hinder evangelism. The hypothetical argument raised against Calvinism is this: "Since God chose His own in eternity past; and, since He grants repentance and faith needed in order to come to Him; and, since all He has chosen will, in fact, come to Him (John 6:37); and all who come to Him are eternally secure; then, it follows that man isn’t involved in salvation." But this is a wrong conclusion. Man is very much involved. God ordains the end—the salvation of lost man. But God also ordains the MEANS to the end— evangelism. God could have ordained any number of ways to communicate salvation. He has given a revelation of Himself in creation and conscience (Romans 1 - 2). But He has specifically chosen to communicate the Gospel message through believers sharing the message of salvation (Romans 10:9-17). So, whether one is a Calvinist or not, evangelism is the responsibility of all believers. Historically, Calvinism not only didn’t diminish the Calvinists’ burden for souls, it purified it! The Calvinists were among the greatest evangelists in the history of the church, motivated by love for their Lord and Savior who chose them and saved them “before the foundation of the world” (Ephesians 1:4).
Before we truly understand the sovereignty of God in salvation, we often think the burden is on us to “produce” decisions for Christ. We act as if a person’s salvation is dependent on us. So when we share the Gospel and it is rejected, we somehow think we failed to talk that person into believing and that we need a more clever or polished presentation of the plan of salvation. We may be tempted to water down the Gospel next time in order to get the desired response. But once we understand the Doctrines of Grace, the pressure to force a “decision” is removed. Now, we witness because we want to be faithful to our dear Lord. Evangelism among Calvinists is driven by the familiar phrase "By His grace, and for His glory!" No, Calvinism shouldn’t hinder evangelism. If anything, it should give our witnessing great boldness with pure motives.