Offering condolences to Christians

Yes, it is permissible to offer them condolences at times of bereavement, to
visit them when they are sick, and to console them when calamity strikes. It
was narrated that Anas (may Allaah be pleased with him) said: “There was a
Jewish boy who used to serve the Prophet (peace and blessings of Allaah
be upon him). He fell sick, and the Prophet (peace and blessings of
Allaah be upon him) came to visit him. He sat at his head and said, ‘Become
Muslim.’ (The boy) looked at his father who was with him, and he (the
father) said to him, ‘Obey Abu’l-Qaasim ( (peace and blessings of Allaah
be upon him)).’ So he became Muslim. Then the Prophet (peace and
blessings of Allaah be upon him) left, saying, ‘Praise be to Allaah who has
saved him from the Fire.’”

(Narrated by al-Bukhaari, 1356) 

And it was narrated from Anas that a Jew invited the Prophet
(peace and blessings of Allaah be upon him) to eat barley bread and
other food, and he accepted the invitation. (Narrated by Ahmad, 13201, with
a saheeh isnaad). 

It should be noted that if a Muslim does that, he should do
so with the intention of calling them to Islam and softening their hearts
towards Islam; he should call them in an appropriate manner and at an
appropriate time. 

It should also be noted that whilst consoling them he should
not pray for forgiveness or mercy or Paradise for their dead, because Allaah
says (interpretation of the meaning): 

“It is not (proper) for the Prophet and those who believe
to ask Allaah’s forgiveness for the Mushrikoon, even though they be of kin”

[al-Tawbah 9:113] 

Rather he should pray for them in a way that is appropriate,
encourage them to be patient and steadfast, and remind them that this is the
way of Allaah with His creation. And Allaah knows best.