What should a Muslim son do for his kaafir father when he dies?
Question

My father has died and he is a kaafir. What are my responsibilities towards him? Should I bury him and arrange his funeral? Should I wash him and shroud him or not? Are there any rights that my father has over me in this respect?
Praise be to Allah.It was narrated in a saheeh hadeeth that Naajiyah ibn Ka’b narrated that ‘Ali ibn Abi Taalib (may Allah be pleased with him) said: “I said to the Prophet (peace and blessings of Allah be upon him), ‘Your old, misguided uncle has died (he was referring to his father Abu Taalib). Who will bury him?’ He said, ‘Go and bury your father.’ [I] said, ‘I will not bury him, for he died as a mushrik.’ He said, ‘Go and bury him, then do not do anything until you come to me.’ So I went and buried him, then I came to him with traces of dust and earth on me. He told me to wash myself, then he made du’aa’ for me in words that were more precious to me than everything on earth.”
(al-Silsilah al-Saheehah, by al-Albaani, no. 161) 
The Shaykh said, commenting on this hadeeth: 
1 – It is allowed for the Muslim to take care of the burial of his mushrik relatives. That does not cancel out his hatred of their shirk. Do you not see that ‘Ali (may Allah be pleased with him) initially refused to bury his father for that very reason, as he said, “he died as a mushrik.” He thought that if he buried him when this was the case, that this was included in the forbidden kind of friendship as referred to in the aayah (interpretation of the meaning): 
“O you who believe! Take not as friends the people who incurred the Wrath of Allah”
[al-Mumtahinah 60:13]
When the Prophet (peace and blessings of Allah be upon him) repeated the command to bury his father, he hastened to obey him and he gave up the notion that had initially occurred to him. This is what obedience means: that a person gives up his opinion in response to the command of his Prophet (peace and blessings of Allah be upon him). It seems to me that a son’s burying his mushrik father or mother is the last act of good companionship that the son can do for his mushrik parent in this world. But after the burial, he cannot make du’aa’ for him or pray for forgiveness for him, because of the unambiguous words of Allah (interpretation of the meaning):
“It is not (proper) for the Prophet and those who believe to ask Allah’s forgiveness for the Mushrikoon, even though they be of kin”
[al-Tawbah 9:113]
If this is the case, then how can people pray for mercy and forgiveness in the pages of newspapers and magazines for some of the kuffaar, in death announcements for money! Let the one who has any concern for his hereafter fear Allah.
2 – It is not prescribed for him to wash or shroud a kaafir, or to offer the funeral prayer over him, even if he was a relative, because the Prophet (peace and blessings of Allah be upon him) did not tell ‘Ali to do that. If that were permissible, he (peace and blessings of Allah be upon him) would have said so, because it is well known that it is not permitted for the Prophet (peace and blessings of Allah be upon him) to delay explaining something at the time when that information is needed. This is the view of the Hanbalis and others. 
3 – It is not prescribed for the relatives of a mushrik to follow his funeral, because the Prophet (peace and blessings of Allah be upon him) did not do that in the case of his uncle, although he was the one who was the most kind and compassionate towards him, and he even prayed to Allah to make his punishment the lightest punishment in Hell. 
In all of that there is a lesson for those who are deceived by pride in their lineage and who do not strive for their Hereafter with their Lord. Allah indeed spoke the truth when He said (interpretation of the meaning):
“there will be no kinship among them that Day, nor will they ask of one another”
[al-Mu’minoon 23:101]
al-Silsilah al-Saheehah by al-Albaani, p. 94