No, couples who are not married should not live together. Even though secular society advocates such sinful behavior, the Bible teaches us that when a man and woman live together, it should be because they're married. The sexual union is for marriage only and is not to be casually had for the sake of pleasure and convenience. Such fornication shows disrespect to each other, motivates people to indulge in sensual pleasure outside of the safety of marriage, risks children out of wedlock, and risks killing children before they are born.
Also, God's Word tells us to avoid the appearance of evil (1 Thess. 5:22). So, even if a couple (unmarried male and female) were to live together and not have sexual relations, the appearance of sinful behavior cannot be avoided. Therefore, because we are to obey God rather than men (Acts 5:29), we are to act in a manner consistent with God's desires for us. Marriage is what allows a man and a woman to live together properly and morally. So, couples who are not married should not live together.