There are at least two things that are exceedingly important for understanding Paul's profound statement concerning the Church in 1 Corinthians 3:16. The first of these is grammatical in nature. In English when we address someone in the second person we use the word "you" whether we are addressing one person or a group. The word "you" is both the singular and plural form of the second person address. Likewise, the form of the verb "to be" does not change from the singular to the plural form. In both instances, we would say "you are". This is not the case in Greek in which there is one pronoun and verb form for addressing a single individual and another for addressing a collective group. The closest parallel we have in English is the southern variation of "ya'll" used to distinguish an address to a group from an individual. In 1 Corinthians 3:16, Paul uses that plural form of the second person Greek pronoun and verb. In other words, Paul is not saying in these verses that each of the individual Corinthians are themselves God's Temple (although he will say something like that in 1 Cor. 6:19). Here Paul is saying, in effect, "Or don't ya'll know that ya'll are the Temple of God in Corinth and that God's Spirit dwells among ya'll?" That is to say, that Paul's statement here is, in fact, a statement about the Church and not about individual believers. The Church is the place where God's Spirit dwells.
The second thing we must recognize in order to appreciate Paul's statement here is the vast significance of the Temple to 1st century Judaism. The Temple in Jerusalem represented the presence of God with the people of Israel. This was God's house. The very place where God's Spirit dwelt. It was the salvation of the Jewish people that God had chosen to make his dwelling among them. It is difficult to overstate just how meaningful the Temple was to Paul and his contemporaries, how much it signified God's presence with his people. This was so much the case in Jeremiah's day that it seems some believed that no harm could possibly befall Jerusalem as long as the Temple still stood because, it was thought, God would not allow his holy dwelling to come to ruin. The Temple = God's presence.
So when Paul tells the Corinthians "you are God's Temple", he is making a bold statement about the Church. Paul is telling the Corinthians that the Spirit of God is no longer limited to dwelling in the Temple. God has actually made his home, his dwelling among the Corinthians themselves. This is the significance of Pentecost; that God's own Holy Spirit has been poured out on all those who place their trust in Jesus Christ. While that Spirit certainly works in each of us individually, here Paul is emphasizing that it is in the Corinthians relationship with one another that God's Spirit dwells and is at work. It is in their coming together as a unified community of faith that they are God's Temple. When we call ourselves the Church we are saying nothing short of claiming that God's Spirit resides among us as a people.
Of course, this serves well the point that Paul has been making all along. He has been telling the Corinthians for three chapters now that they must be unified rather than fragmenting themselves into different groups centered around different leaders. Here, Paul reminds them they are all together the Temple of God. Therefore, if they destroy their relationships with one another then they are destroying God's Temple. Paul is telling them that their lack of unity is actually destroying the very place where God's Spirit wishes to make his dwelling. The Corinthians are destroying the house of God by destroying their relationships with one another. This means that disunity in the Church is more than people just not getting along. It is a driving out of the Spirit of God; the very thing that makes us who we are as God's people. It is a failure to be the Church at all.
Paul finishes this section by reminding the Corinthians (again!) that human wisdom is not God's wisdom. Therefore, they should not boast in men (or human leadership). And he goes on to say "For all things are yours, whether Paul or Apollos or Cephas or the world or life or death or the present of the future- all are yours, and you are Christ's and Christ is God's". I think what Paul is essentially saying to the Corinthians here is that they have taken to small and narrow a view of things. Here the Corinthians are striving after mere men, mere human leaders and their positions of honor and authority and Paul is saying "Why would you strive after those things. As followers of Christ, as the Temple of God, everything already belongs to you. You have already been given everything you could ever ask for. God's very own Spirit has made you his home. The Spirit of the almighty God and creator of the universe is at work in your relationships with one another and you want to strive after merely human positions of honor? God has given himself to you and you want to trade that for human leaders who are nothing but servants of that very same God!"
But, of course, this is what we continue to do. God has given himself to us, made his own Spirit to reside among us, and all we can seem to think about is "if we just had a little more money... if we just had a little higher attendance...if we just had a few more young able bodies to do the work...if we just had....". We have the Spirit of God among us! What else do we need to do the work God has entrusted to us when God has entrusted his very self to us? Paul encourages us to take a larger view of things; encourages us not to dream so small that we go after these merely human things but instead to remember that we are the Temple of God and that in itself is a truly miraculous things. God's Spirit has made his dwelling among us.