I don't remember exactly when I found out there were Christians who don't celebrate Christmas, but I remember how surprised I was. Really? There are actually Christians who don't celebrate Jesus' birthday? As it turns out, yes. Not only are there Christians who do not celebrate Christmas, but those who actively condemn those who do. I even remember one woman who said that Christmas is actually a combination of Christ and Mass (which is true), the latter meaning a rejection of Christ, and thus if you're celebrating Christmas, you're actually rejecting Christ. She even said if I felt this was a judgemental statement (which, of course, it is), that was my own conscience telling me that statement was true.
But here's what the Bible says: There are no wrong answers here. There are many things the Bible clearly says we must do. Celebrating Christmas isn't one of them. There are many things the Bible clearly says we mustn't do. Again, celebrating Christmas isn't one of them. Christians are free to celebrate, or not celebrate, any festival we want, as they're all just a shadow of things to come (Colossians 2:16-17). If you want to dance in the shadow until the real thing comes, go for it. If you want nothing to do with the shadow and just want to wait for the real thing, that's your call.
Romans 14 gives us the method by which we make that decision. In it, Paul lays out the difference between the "weaker" and "stronger" brethren. The stronger brethren have a more liberal view about what Paul calls "doubtful disputations". They are fully convinced that they may eat, drink and celebrate anything they want. Weaker brethren take a more restrained view. They believe certain foods, drinks or celebrations are off limits. No, I can't eat bacon. No, my wine must not contain alcohol. No, I mustn't celebrate Christmas. And Paul lays out both views, those of the stronger brethren, and those of the weaker, as being entirely valid before God.
But Paul's letter isn't designed to just say "meh, do what you like. I don't care." There are two important points. The first is that we are not to divide over this kind of thing. The stronger brethren must never look down on the weaker, nor must the weaker condemn the stronger. Each is to treat the other with the same love and respect as they would if they were fully agreed. He even goes a step further in exhorting the stronger brothers to, at least temporarily, forfeit their own liberty for the sake of the weaker brother, as in verse 15, he says that if your brother is grieved by your food, you're not loving him, but destroying him. (Keep in mind that although food receives more focus, the entire passage includes food, drink, and special occasions almost interchangeably).
Furthermore, in 1 Corinthians 8:13, Paul even says he would never eat meat again if such a sacrifice would prevent his brother from stumbling. In context, Paul is dealing with Christians who, though they have accepted Christ and turned from their idols, still don't fully understand that those idols are not even conscious. Thus, when food is offered to those idols, they still see it as food offered to a god, and avoid it, so when they see a Christian, who knows it isn't a god, eating that food, they see it as a Christian worshipping that god, and so they become tempted or offended.
You may remember I mentioned above, the woman who said Christmas is the rejection of Christ, because Mass is a Catholic worship ceremony. She is not an ex-Catholic, but if she was, that could be a comparable scenario. If a Catholic comes out of Catholicism, yet sees Christians still celebrating Christmas, they could associate that with their celebrations of Christmas while still Catholic, and may bring all of their Catholic traditions to it. That might cause them to stumble. And thus, it is fair to say that it is a good idea, if you know a Christian to whom Christmas had a specific religious meaning before their conversion, that perhaps you should avoid Christmas in their presence.
But Paul has a more important point to make as well. The point that although this functions as a personal choice, it's really not. As Christians, we are not supposed to be totally independent. We belong to God. Thus, every decision we make, whether it's eating specific foods, drinking specific drinks or celebrating specific occasions, must be made according to what we believe He would want us to do. This means that even stronger brethren may actually be weaker brethren.
I am a rather interesting example. Morally speaking, I am a stronger brother. I firmly believe that meat is permissible. It's kind of hard to deny that, especially when Jesus was rather helpful to His disciples when they tried to catch some fish. I do not even believe it would be immoral if I went out right now and ordered a KFC boneless banquet all for myself (unless you consider that gluttony, but that's not the point I'm making here). Yet, there is an element of "weaker" brother in me in that I hate what meat is. Meat is the broken remains of what was once a beautiful living creature. A creature that desired to live, and probably even trusted the one who killed it with its life. I've personally looked after animals that would eventually end up on someone's plate, and so although I believe it is morally acceptable, it is mentally impossible for me to not see the tragedy of it all. Thus, under normal circumstances, I prefer a vegetarian lifestyle, and it could be argued that it would be a sin for me to eat meat, simply because I am not fully convinced in my own mind that it is a God honoring thing to do. By contrast, when I am sick, or if (God forbid) I ever find myself in a survival situation in which eating animals would be essential for my own survival, I would be fully convinced, because I know that a human, such as myself, is significantly more valuable than an animal.
All of the above shows that my initial surprise that some Christians don't celebrate Christmas is, to put it bluntly, somewhat arrogant. As a stronger brother, I am free to celebrate Christmas, and as weaker brethren, they are free to abstain. Stronger brethren should respect weaker brethren, and weaker brethren should likewise respect stronger brethren. In the end, we all worship the same God, and it is far better to unite under Him despite our differences than to divide under Him and experience some form of loss when He returns.