No, a Christian should not date or marry an unbeliever:
"Do not be bound together with unbelievers; for what partnership have righteousness and lawlessness, or what fellowship has light with darkness? 15 Or what harmony has Christ with Belial, or what has a believer in common with an unbeliever?" (2 Cor. 6:14-15).
Unfortunately, some Christians hope that they can convert the person he/she is dating or marrying. They often think that their spirituality is strong enough so that they can witness, or motivate the other person to convert through their patience and love. Why do they think this? Three reasons: They love the person and are emotionally blinded, they are naive, or they do not know God's Word.
This may sound harsh, but something as serious as marrying an unbeliever needs to be dealt with properly and to-the-point. We are not to compromise the will of God and endanger our spiritual well-being. All we need to do is look in the Old Testament to see why God says not to marry unbelievers. This is what God said to the Israelites.
"When the Lord your God shall bring you into the land where you are entering to possess it, and shall clear away many nations before you, the Hittites and the Girgashites and the Amorites and the Canaanites and the Perizzites and the Hivites and the Jebusites, seven nations greater and stronger than you, 2 and when the Lord your God shall deliver them before you, and you shall defeat them, then you shall utterly destroy them. You shall make no covenant with them and show no favor to them. 3 Furthermore, you shall not intermarry with them; you shall not give your daughters to their sons, nor shall you take their daughters for your sons. 4 For they will turn your sons away from following Me to serve other gods," (Deut. 7:1-4).
The reason not to marry an unbeliever is that he/she can influence you away from the Lord. Don't be so naive to think that "you" will never fall. It happens all the time.
If you marry an unbeliever and have children, how will it effect their spirituality to have the parents divided over spiritual things? Is it a help or a hindrance to their spiritual health? Obviously, it is a hindrance.
Unfortunately, too many people do not take into account the extremely serious situation of children and their eternal destiny. Yet, because of "love" and because they listen to their hearts over the Word of God, many people marry unbelievers anyway... and often suffer dire consequences.
What if you are already married to an unbeliever?
If you are already married to an unbeliever, then you need to love him/her as best you can, set a good example of being a Christian, and pray for your spouse's salvation. You cannot leave your spouse (unless there is physical abuse, adultery, or if you are abandoned). You need to stick it out. Seek the Lord.