Let's start by taking the "birth" day. It is commonly accepted by scholars that the day Jesus was born was any day but December 25. The shepherds were in the fields and wouldn't have been in December. Most scholars believe Jesus was born sometime late October or early November. The Bible gives no reference to the day, so that should imply at least to some degree the actual day was not important; otherwise, the Holy Spirit would have made sure that information was included in the Bible. Also, there is no specific reference to any celebrating Christ's birth in the Bible in a recurring sense, and there is no historical evidence that it was celebrated by the early Church. So, we have to admit that at a minimum, celebrating Jesus' birth is not instituted by God, that the idea of celebrating it is man-made, and that saying this is Jesus' birthday is not based in the truth.
Which leads to the question: where, when and by whom was this celebration instituted? The Pope in the 3rd century-ish instituted the celebrating of it. The holiday is named after the Mass of Christ, which is a Catholic rite whereby Christ is essentially re-crucified for the sins of those participating, and where the bread and wine supposedly are turned into the actual body and blood of Christ by the priest, which the people take into themselves. This rite is, and has been classified by greats such as John Knox, as an abomination to God, in that, Christ died once for all, and the Mass flies in the face of that and blasphemes the work of Christ on the cross. Also, the supposed transubstantiation is not what the Bible says about the bread and wine –- they are symbols. So, the Christmas holiday is named after that.
Why the 25th of December? The winter solstice is around that time, and for centuries, pagans celebrated the solstice using the various accoutrements commonly known in Christmas celebrations today. Without getting into each of them, just about every single thing common to Christmas (the tree, holly, gift-giving, etc.) is pagan in origin. In fact, some of the pagan celebrations around that time celebrate a mother-child relationship regarding their pagan gods.
Well, the Pope thought it would be good to try to Christianize that day, to witness to folks, and to get them to focus on God instead of their pagan gods while still being allowed to celebrate. This is syncretization to the core –- to attempt to unite and harmonize especially without critical examination or logical unity. It may have seemed logical at the time, but what does the Bible say about syncretizing with pagans or their accoutrements? Time after time you see in the Bible that God says 1) do not do it because always you will become like them and not the other way around -- Solomon learned this the hard way and lost his kingdom because of it; and 2) whenever the Israelites went into a pagan land, the Lord never said to take the religious things that were there and use them for Him -- He always had them destroy everything completely -- all the buildings and anything related to their paganism. God also says to be separate, holy and not conformed to this world. The joining with the pagans on that day, even to celebrate it supposedly in honor of Christ, is a joining and a conforming to the world.
If you study the Bible closely, you will see that God institutes how He will be worshipped, and expects that and nothing else. We are to obey what He says, and just because He doesn't reference something directly doesn't mean we can just go ahead and do it, even if we think it is honoring to God. There are many references in the Bible where people tried to sincerely honor God in their own way, and God rejected those acts, whether it was Cain and the fruit of his labors, or Aaron’s sons who brought their strange fire to God and were struck down dead, to David and his cart for the ark. God never said, "David, don't use a cart." He said, have the priests carry it with the rods. David did it his own way, and the result was bad. If we were allowed to do anything that God has not specifically forbidden, the Bible would be endless with tons of "do nots" to account for every machination our minds could conceive. That's why it's important that the Bible stand alone as the only way directing behavior and action, especially in worship to God.
If not letting the Bible be our guide in worship is taken out to its logical conclusion, then anything I do supposedly becomes honoring to God just because I say it is for Him, which means that worship is left up to my imagination. Eventually I might be jumping for the Lord, standing on my hands for the Lord, crossing my eyes for the Lord (because it keeps me from focusing on the things of this world!), etc., etc. There’s no end to where that can go if we do not simply take the Bible as the limiter of our actions. Today's church is another reflection of this – their idea is that if they think it's a good idea and they say it's honoring to God, then it must be. God also says to be wary of following the tradition of men, which the Christmas holiday certainly is.
Let's also look at how it's celebrated. There is often over eating and drinking, which shows a lack of temperance, part of the fruit of the Spirit. Also, do children really think of Christmas as a time of giving, or a time of getting? And what of all of the commercialization, even using the name of Jesus for monetary gain? Further, the entire world loves Christmas, but hates Sundays (except for Christmas and Easter). Why? If Christmas was truly of the Lord, would not the world hate it?
Still, one might ask, "Well, it's a good opportunity to witness or be an example to others." If you want to be an example, don't celebrate it and see how much opportunity you have to speak of the Lord Jesus. Also, which actually shows being different (as Christians are supposed to be) –- celebrating it or not? If a Christian is living a truly Christian life, he will have plenty of other opportunities to speak of Christ, because his life will be so different at all times.
Finally, I'm not saying don't recognize the birth of Christ –- his incarnation was a precursor to Him dying on the cross and making atonement, and was a great condescension by the Lord God Almighty. However, the Lord has given (by institution in the Bible) 52 holy days a year to celebrate Christ; and not only that, we have 364 other days to do this as well, which we should, as Christ is the reason we live and have hope. The supposed "spirit" of Christmas should be in us every day of the year and should be exemplified in our lives in charity (love) to others. And what better way to be a witness or have opportunity to speak of Christ than through our actions each and every day?
Sources and for further study (I do not necessary vouch for other content on these Web sites):
http://web.archive.org/web/20121206161322/http://reformedonline.com/view/reformedonline/xmas.htm (this is a long one, but detailed)
https://web.archive.org/web/20160203141803/http://www.spurgeon.org/sermons/1026.htm (first paragraph)