What Scripture supports the doctrine of the "communion of saints"?

In the Apostles' Creed, the faithful confess belief in the "communion of saints." The Catechism of the Catholic Church teaches that the Apostles' Creed "is rightly considered to be a faithful summary of the apostles' faith. It is the ancient baptismal symbol of the Church of Rome. Its great authority arises from this fact: it is 'the Creed of the Roman Church, the See of Peter, the first of the apostles, to which he brought the common faith'" (no. 194, quoting St. Ambrose, a 4th-century bishop and doctor of the Church).

We start there because the communion of saints is part of the "deposit of faith" that was entrusted to the Church. While it's most important to be provide scriptural bases for Church teachings when it comes to apologetics and ecumenism, at the same time we realize that Catholics do not hold to the "sola scriptura" theory that was developed by 16th-century Protestantism, and thus our inquiry is not limited to the sacred page outside the larger context of the Church.

But since you asked, let's briefly examine some of the biblical foundations.

We can say that the communion of saints is based on four rock-solid, scriptural points:

(1) All Christians are members of Christ’s body and one another (Romans 12:5; 1 Corinthians 12:27).

(2) Jesus has only one body (Ephesians 4:4; Colossians 3:15). (There are countless passages that identify the Church as the "Body of Christ.")

(3) Death cannot separate Christians from Christ or from one another (Romans 8:35-39; see also Luke 20:37-38).

(4) Christians are bound in mutual love (John 13:34-35; Romans 12:10).

In John 15:1-5, Jesus tells us that He is the vine and we are the branches. As branches are connected to a vine, participating in a single life, we are connected to Jesus and to one another in His Mystical Body. Saint Paul speaks many times about the importance of this unity in one body (e.g., 1 Corinthians 12:12-27; Romans 12:4-16). As members of the same family, we are able to pray for one another and to ask others—including the saints in heaven—to pray for us. Nothing, not even death, can break that union between Christ and His body and the members with one another (cf. Romans 8:35-39). Everyone who is “in Christ” (2 Corinthians 5:17) participates in His life not only on earth, but even more fully in the glory of heaven.

The foregoing is taken from Suprenant and Gray, Faith Facts: Answers to Catholic Questions, vol. 1. We recommend the entire chapter of that book dealing with "communion of saints" for more in-depth study of this subject. 

Also, the treatment of the communion of saints in the Catechism of the Catholic Church (nos. 946-62) contains many references to Scripture as well as Vatican II and other traditional sources.