Praise be to Allah.
What is forbidden with regard to congratulating the Christians on the occasion of their festivals is expressing joy to them, being overly polite and showing approval of their actions, even if that is only expressed outwardly without feeling it inwardly.
The prohibition concerns those who show any kind of participation or approval, such as giving gifts, verbal greetings, taking time off work, making food, going to places of leisure, and other customs of celebration. Having any intention other than what is said (when congratulating them) does not make it permissible. The outward appearance of these actions is sufficient to say that it is prohibited.
It is well known that most of those who take a lenient attitude towards such matters do not intend to join the Christians in their shirk; rather what motivates them in some cases is showing politeness, and in other cases it is shyness, but politeness with regard to falsehood is not permissible; rather what is required is to denounce the evil and strive to change it.
Shaykh al-Islam Ibn Taymiyah (may Allah have mercy on him) said in Majmoo‘ al-Fataawa (2/488):
It is not permissible for the Muslims to imitate them in any way that is unique to their festivals, whether it be food, clothes, bathing, lighting fires or refraining from usual work or worship, and so on. And it is not permissible to give a feast or to exchange gifts or to sell things that help them to celebrate their festivals, or to let children and others play the games that are played on their festivals, or to adorn oneself or put up decorations.
In general, (Muslims) are not allowed to single out the festivals of the kuffaar for any of these rituals or customs. Rather the day of their festivals is just an ordinary day for the Muslims, and they should not single it out for any activity that is part of what the kuffaar do on these days.
As for what the Muslims do deliberately, that was regarded as makrooh by a number of the earlier and later generations. As for singling out (such days) for the things mentioned above, there is no dispute concerning that among the scholars; in fact some of the scholars are of the view that the one who does these things is a kaafir, because it involves venerating the symbols of kufr. And some of them said that the one who slaughters a sheep on the day of their festival, it is as if he slaughtered a pig.
‘Abdullah ibn ‘Amr ibn al-‘Aas said: The one who follows the tradition of non-Muslims and observes their Nawrooz and Mahrajaan (festivals), and imitates them until he dies like that, will be gathered with them on the Day of Resurrection.
Ameer al-Mu’mineen ‘Umar ibn al-Khattaab, the Sahaabah and all the imams of the Muslims stipulated that they should not celebrate their festivals openly in the lands of the Muslims; rather they should do that privately in their houses.
One of the early generations said concerning the verse in which Allah, may He be exalted, says (interpretation of the meaning): “And those who do not witness falsehood” [al-Furqaan 25:72]: (This refers to) the festivals of the kuffaar; if that has to do with witnessing them, and not actually doing anything, then how about doing things that are unique to those festivals? It was narrated from the Prophet (blessings and peace of Allah be upon him) in al-Musnad and as-Sunan that he said: “Whoever imitates a people is one of them.” According to another version he said: “He is not one of us who imitates others.” This is a jayyid isnaad. If this is the case with regard to imitating them, and if this has to do with customs, then how about imitating them in matters that are more serious than that? The majority of imams regard it as makrooh, either in the sense of it being prohibited or in the sense of it being discouraged, to eat meat that they slaughter at the time of their festivals and sacrifices, and they include that under the heading of that which is sacrificed for something other than Allah and that which has been sacrificed on stone altars (for idols). Similarly they also forbade helping them in their festivals by giving gifts or selling to them, and they said: It is not permissible for the Muslims to sell to the Christians anything for the purpose of their festivals, whether that be meat, blood or garments; they should not hire out any mounts to them or help them with any of their religious affairs, because that comes under the heading of venerating their shirk and helping them in their kufr. Those in positions of authority should forbid the Muslims to do that, because Allah, may He be exalted, says (interpretation of the meaning): “Help you one another in Al‑Birr and At‑Taqwa (virtue, righteousness and piety); but do not help one another in sin and transgression”[al-Maa’idah 5:2]. Moreover, it is not permissible for the Muslim to help them in drinking alcohol by pressing the juice and so on, so how about helping in that which is one of the symbols of kufr?! If it is not permissible for him to help them himself, then how about if he is the one who is actually doing that?! End quote.
And Allah knows best.