At least three times in the last year, I have heard from Christians who are concerned that they shouldn't have a Christmas tree, or maybe shouldn't celebrate Christmas, because many Christmas customs are adopted from pagan customs. Some Christians worry that the date of Christmas was picked to replace pagan holidays such as Saturnalia (likely true), that the Christmas tree came from pagan rituals (uncertain, but possible), or that the star on the tree comes from star worship (not too likely, in my opinion).
To answer this concern, let me ask a very different question: is it OK for Christians to use the names for the days of the week and the months? The days of the week honor Norse gods: Sun day, Moon day, Tiw's day, Odin's day, Thor's day, Freya's day, Saturn day. Some of the months honor Greek and Roman gods (Janus, Mars, Aphrodite, Maia, Juno) while others honor deified Roman emperors (Julius, Augustus). But no one seems to have a problem with this, for a very good reason: when Christians say "Thursday," they are not worshipping Thor, and no one else thinks they are, either. The Bible prohibits idolatry, but idolatry consists of worshipping other gods. Worship is not something that one does unintentionally; it's a matter of the heart.
Symbols and words are not permanently tainted merely because they are used by another religion ("I know and am convinced in the Lord that nothing is unclean in itself..." says Paul in Rom 14:14, referring to meat sacrificed to idols). Cults misuse baptism and the Lord's Supper, but that does not mean we should no longer baptize or share in communion. When Solomon built the first temple for God in the Old Testament, there had already been pagan temples for at least two thousand years. The design of Solomon's temple even had some similarities to Egyptian temples. The fact that temples were used by other religions did not make it wrong to build a temple to God or to use the temple as a metaphor for God's people. There are lots of other examples in the Bible of using (or redeeming) terms and symbols from the pagan world. Paul quotes the poem "A Hymn to Zeus" in Acts 17:28. While astrology and the worship of stars is condemned in the Bible, stars are repeatedly used as symbols of Jesus, Israel, and the church - so it is OK for you to put a star on your Christmas tree.
Finally, celebrating Christmas is allowable because the Scripture gives us personal freedom in such matters. In 1 Cor 8-10, Paul tells Christians not to participate in idol feasts, because idol feasts involved acts of worship to other gods, and everyone knew such feasts were designed to honor other gods. But in 1 Cor 10:25-33, Paul says that Christians can eat meat that was sold in an idol market, because it is not an act of worship, and the meat is not permanently tainted. Paul also says that Christians have freedom to celebrate religious holidays or not according to their own conscience, and even warns us not to condemn others for their decisions in these areas of freedom (Rom 14:1-14, Col 2:16-17). The Bible forbids worship of other gods - but celebrating Christmas by putting up a Christmas tree does not constitute worship, especially when we do it in honor of Jesus.
"Do not let what is for you a good thing be spoken of as evil, for the kingdom of God is not eating and drinking [or celebrating Christmas or not!], but righteousness and peace and joy in the Holy Spirit" (Rom 14:16-17). The Christian life is primarily about living out the virtues of Christ, empowered by the Holy Spirit. If Christmas is a good thing for you that leads you to think about Jesus, then enjoy it - but more importantly, make sure that your Christian life is primarily about things that really matter, not minor quibbles about Christmas.
The picture: The Nativity, by an unknown Ottonian, ca. 1025-1050.