My government is mandating that everyone receive the COVID-19 vaccine, but I heard some Christians are refusing to take it. Should a Christian obey the government mandate or can we refuse to take the vaccine?
The Bible is silent on the topic of medical treatments, which means we must apply a basic principle for Christian living called "liberty." All Christians are free to decide for themselves in matters not addressed specifically by Scripture so long as our choices do not violate Scripture nor our conscience. The principle of liberty sets an important standard for Christian fellowship that believers respect and honor the choices made by other believers in matters outside the Bible.
Since the Bible does not set requirements on medical treatments (including taking vaccines), Christians have liberty to decide the matter for themselves, and we must leave room for different choices within the body of Christ. One believer may elect to receive a vaccine while another believer may refuse the same. Nevertheless, both are acting properly because each is following his or her own conscience in the spirit of Christian liberty.
On the other hand, Scripture teaches that believers do not have the liberty to disobey the law. When matters of liberty come into conflict with government directives, a believer must prioritize obeying the government ahead of exercising personal liberty, for to do otherwise is sin, as Paul says:
Rom. 13:1 Every person is to be in subjection to the governing authorities. For there is no authority except from God, and those which exist are established by God.
Rom. 13:2 Therefore whoever resists authority has opposed the ordinance of God; and they who have opposed will receive condemnation upon themselves.
Scripture sets a very high standard for our submission to authority. Paul says all governments are agents of God established to do His will, so in effect, Paul is saying that to disobey the government is to disobey God Himself.
Can a believer refuse a government's order to take the vaccine on the basis of personal conscience? For example, if we suspect a vaccine was developed in immoral ways (i.e., using aborted fetal tissue, etc.), can we refuse the vaccine on moral grounds?
Yes, a believer may refuse to obey the government on the basis of conscience, however, the believer is duty-bound in such circumstances to accomplish proper due diligence before deciding the matter.
The question of obedience to government vs. obedience to God is addressed as part of our study on Romans in Chapter 12-15. In these chapters, Paul gives us a hierarchy for obedience, including a system for determining how we should obey in cases where our duty to God and our duty to other authorities come into conflict. We highly recommend you listen to the entire Romans Bible study, since it is a transformative study for every Christian.