What is the Biblical basis for the Holy Spirit being God? I know that Trinitarians believe that God exists as one God in three persons: God the Father, God the Son, and God the Holy Spirit.
That God the Father is God is patently obvious from the entire Old Testament.
That God the Son is God is made forcefully in John, Colossians, and throughout the New Testament.
We also know that Jesus sent the Holy Spirit, but...
Where in the Bible does it say that the Holy Spirit is God?
In Acts 5:1-4 it says Ananias has lied to the Holy Spirit and has therefore lied to God:


  Now a man named Ananias, together with his wife Sapphira, also sold a
  piece of property. With his wife’s full knowledge he kept back part
  of the money for himself, but brought the rest and put it at the
  apostles’ feet.
  Then Peter said, "Ananias, how is it that Satan has so filled your
  heart that you have lied to the Holy Spirit and have kept for yourself
  some of the money you received for the land? Didn’t it belong to you
  before it was sold? And after it was sold, wasn’t the money at your
  disposal? What made you think of doing such a thing? You have not lied
  just to human beings but to God."


In 2 Corinthians 3:17-18 we see the Spirit is the Lord:


  Now the Lord is the Spirit, and where the Spirit of the Lord is, there
  is freedom. 18 And we all, who with unveiled faces contemplate[a] the
  Lord’s glory, are being transformed into his image with
  ever-increasing glory, which comes from the Lord, who is the Spirit.


The bold text is my emphasis of course.