Praise be to Allah.
Shaykh Ibn `Uthaymin (may Allah have mercy on him) said:
Greeting the non-Muslims on Christmas and other religious holidays of theirs is prohibited by consensus.
Ibn Al-Qayyim (may Allah have mercy on him) said in Ahkam Ahl Adh-Dhimmah:
"Congratulating the non-Muslims on the rituals that belong only to them is prohibited by consensus, as is congratulating them on their festivals and fasts by saying ‘A happy festival to you’ or ‘May you enjoy your festival ,’ and so on. If the one who says this has been saved from [falling into] disbelief, it is still forbidden. It is like congratulating someone for prostrating to the cross, or even worse than that.
It is as great a sin as congratulating someone for drinking wine, or murdering someone, or having illicit sexual relations, and so on. Many of those who have no respect for their religion fall into this error; they do not realise the offensiveness of their actions. Whoever congratulates a person for his disobedience or Bbid`ah or disbelief exposes himself to the wrath and anger of Allah."
Congratulating the non-Muslims on their religious festivals is prohibited to the extent described by Ibn Al-Qayyim (may Allah have mercy on him) because it implies that one accepts or approves of their rituals of disbelief, even if one would not accept those things for oneself. But the Muslim should not accept the rituals of disbelief or congratulate anyone else for them, because Allah does not accept any of that at all, as He says (interpretation of the meaning):
"If you disbelieve, then verily, Allah is not in need of you, He likes not disbelief for His slaves. And if you are grateful (by being believers), He is pleased therewith for you. . ." [Az-Zumar 39:7]
". . . This day, I have perfected your religion for you, completed My favour upon you, and have chosen for you Islam as your religion . . ." [Al-Ma’idah 5:3]
So congratulating them is forbidden, whether they are one’s colleagues at work or otherwise.
If they greet us on the occasion of their festivals, we should not respond, because these are not our festivals, and because they are not festivals which are acceptable to Allah.
These festivals are innovations in their religions, and even those which may have been prescribed formerly have been abrogated by the religion of Islam, with which Allah sent Muhammad (peace and blessings of Allah be upon him) to the whole of mankind. Allah says (interpretation of the meaning):
"Whoever seeks a religion other than Islam, it will never be accepted of him, and in the Hereafter he will be one of the losers." [Al `Imran 3:85]
It is prohibited for a Muslim to accept invitations on such occasions, because this is worse than congratulating them as it implies taking part in their celebrations .
Similarly, Muslims are forbidden to imitate the non-Muslims by having parties on such occasions, or exchanging gifts , or giving out sweets or food, or taking time off work, etc. The Prophet (peace and blessings of Allah be upon him) said: "Whoever imitates a people is one of them."
Shaykh Al-Islam Ibn Taymiyah (may Allah have mercy on him) said in his book Iqtida’ As-Siraat Al-Mustaqim Mukhalifat ashab Al-jahim:
"Imitating them in some of their festivals implies that one is pleased with their false beliefs and practices, and gives them the hope that they may have the opportunity to humiliate and mislead the weak [Muslims]."
Whoever does anything of this sort is a sinner, whether he does it out of politeness or to be friendly, or because he is too shy to refuse, or for whatever other reason, because this is hypocrisy in Islam, and because it makes the non-Muslims feel proud of their religion.
Allah is the One Whom we ask to make the Muslims feel proud of their religion, to help them adhere steadfastly to it, and to make them victorious over their enemies, for He is the Strong and Omnipotent.” (Majmu` Fatawa wa Rasa’il Ash-Shaykh Ibn `Uthaymin, 3/369)
And Allah knows best.