A Jewish man converts to Christianity and becomes estranged from his family. When he dies, his next-of-kin is his brother. Is the brother obligated to bury the deceased as a Christian, or can he bury him as a Jew (which he never ceased to be)? Does it make a difference whether the deceased left instructions to be buried as a Christian?
According to this article describing the burial of a Jewish police informer in Luban, Russia (who had presumably caused much suffering on the part of his fellow Jews), Rav Moshe Feinstein was of the opinion that "after death according to Jewish law a person doesn’t own his body and cannot leave orders about his body for after death. Therefore I say you need to listen to Jewish law and bury this man in the way permitted by Jewish law.”
When the burial society which made the shayla objected to this, he continued, “It’s our job to follow the law and my job as rabbi is to make sure that the law is indeed kept. He must be buried according to Jewish law. As for his sins, he will be judged in heaven and he will get forgiveness according to his judgment. It is none of our concern.”
This would seem to indicate that we are obligated to provide even a person who "converted" to another religion with a proper, halachic Jewish burial