It is certainly true that "the heaven and heaven of heavens" cannot contain God. He is omnipresent. There is no realm to which His presence does not reach. The psalmist, exalting God's omnipresence, said, "If I make my bed in hell, behold, You are there" (Ps. 139:8).
So to say that God dwells in heaven is not to say that He is contained there. But it is uniquely His home, His center of operations, His command post. It is the place where His throne resides. And it is where the most perfect worship of Him occurs. It is in that sense that we say heaven is His dwelling place.
In fact, heaven is so closely identified with God in the Jewish conception that it actually became a euphemism for God Himself. People fearful of taking the Lord's name in vain substituted Heaven for the name of God. Particularly during the Inter-testamental Period (the 400 years between the events of the Old Testament and those of the New), the Jewish people developed an almost superstitious fear of using God's name. They believed the covenant name of God (Yahweh or Jehovah) was too holy to pass through human lips. So they began substituting other terms in place of God's name, and "heaven" became a common substitute.
By New Testament times that practice was so ingrained that the Jewish people understood most references to heaven as references to God Himself. Luke refers to "the Kingdom of God." But Matthew, writing to a predominantly Jewish readership, calls it "the kingdom of heaven" (cf. Luke 8:10; Matthew 13:11). We see another example of the use of heaven as a euphemism for God in Luke 15:18, for example, where the prodigal son, rehearsing what he would say to his father, says, "I will arise and go to my father, and will say to him, 'Father, I have sinned against heaven and before you." He meant, of course, that he had sinned against God.
Although heaven is often used this way in place of God's name, we must not conclude that Scripture intends to equate heaven with God Himself. The terms are not synonyms. God transcends heaven. Heaven, in the end, is a place--the place where God dwells, the place where the elect will dwell with Him for all eternity, the heaven of heavens, the third heaven.
Adapted from John MacArthur, The Glory of Heaven (Wheaton, IL: Crossway, 1996), 55-59.