Talking from a Shia perspective, if it is a tradition of our society then there is no problem to celebrate it as long as no Haraam action is performed during the celebration. We have a similar celebration called Nowruz which is not Islamic and celebrates the beginning of spring. [Actually there are some Ahadeeth which demonstrates the religious importance of Nowruz but there is a debate between the scholars if such Ahadeeth are weak or not.] It is considered not only Halal but also encouraged as the family members gather together, children are happy for the vacation and travelings, gifts strength the family ties, and so on. It is not against any rules of Allah but also goes well with them so why should it be Haraam. There are also several other non-Islamic celebrations in Iran, like the beginning of Winter which is the longest night of the year and is called the Yalda Night. According to Wikipedia:
Following the fall of the Sassanid Empire and the subsequent rise of
Islam in Persia/Iran, the religious significance of the event was
lost, and like other Zoroastrian festivals, Yalda became a social
occasion when family and close friends would get together.
that is Islam has revised the old traditions, keeping their good parts and omitting the Kufr rituals and shirk points in them.
We are recommended to follow the traditions of the society that we live in, and even their common style of life, but only in those respects that they conflict with Islam then we should avoid them. Even you can notice that no two culture in Islamic cultures are exactly the same, each have their own traditions and life style but hopefully they all stick to the rules of Allah.
About the Christmas, if it is a mere secular tradition as far as I know it wouldn't be Haraam to participate it as long as it is good on its own, you can change your intention to do all such good deeds for the sake of Allah. However, if it is to celebrate the birthday of Jesus the Christ --peace be upon him--, then it should be better to participate the celebration, while not committing any sin during the celebration. We similarly celebrate all the birthdays of Imams and prophets (that we know their birthday, I only remember Ibraheem --peace be upon him-- whose birthday is recorded as 1st of dhi-Hadjeh) as the day is a day that Allah has gifted us with such a blessing.
Only one point remains to say here. As the Lunar Hijri years are shorter in length than Iranians and Gregorian years for about 11 days, the Lunar years move with respect to those Iranian and Gregorian calenders and it is possible that a religious mourning day or week or etc. (like Ashura or Muharram) coincides with a same day that is traditionally going to be celebrated, like Christmas or Nowruz. Only when such occasions occur we prefer the religious event on the traditional event. So for example we never celebrate Nowruz when it falls in the first 10 days of Muharram.
And God knows best.
EDIT. About why celebrating personal (like birthdays of the family members and friends), cultural (like the beginning of spring or a new year) and national (like 22nd of Bahman in Iran, the day Islamic revolution won against the American Backed Regime of Shah) celebrations is not Bid'ah (innovation), one should note that innovation is adding a part to Islam when that part doesn't indeed belong to Islam, or equivalently remove a part from Islam when it is indeed a part of Islam. However, none of such celebrations are claimed to be Islamic, that is, parts of Islam, as if they were then all Muslims around the world in all the times were requested to do so and of course it is not the case, not anyone except my family will celebrate my birthday not anyone from one culture is required to celebrate a cultural celebration of another culture. However, difference of cultures may be viewed as difference of languages and color of the bodies which are mentioned in Quran as:
وَمِنْ آيَاتِهِ خَلْقُ السَّمَاوَاتِ وَالْأَرْضِ وَاخْتِلَافُ
أَلْسِنَتِكُمْ وَأَلْوَانِكُمْ إِنَّ فِي ذَٰلِكَ لَآيَاتٍ
And among His Signs is the creation of the heavens and the earth, and
the variations in your languages and your colors: verily in that are
Signs for those who know. [30:22]
PS. Several years ago I personally decided to get rid of all the cultural and National dependencies and stick merely to religion. I saw how much difficulties my decision brought to my family for them to adapt. The bad point was that it was only my own desire to be so and couldn't find any Islamic evidence to back my decision with. Then after sometimes of bothering my family with such an extreme decision, repeatedly explaining to the others why I cannot behave like them such that it almost drew everyone's attention toward me and my beliefs and decisions, I found evidences in (Shia) Islam that what is correct (and difficult as well) is to live like everyone but at the same time to be submitted to Allah,