Does Torah permit having friends who are Christians/Muslims/other forms of Avodah Zarah?
The Talmud says that if a non-Jew is destitute, we must provide charity; if ill, we tend to them; if they die without relatives, we bury them.
This applies even to downright pagans, and certainly to non-Jews who are not pagans. (Muslims definitely fall into the latter category.)
There are concerns of becoming overly chummy with non-Jews as it can lead to intermarriage; Rabbi Aaron Soloveichik remarked that the ban here was on "fellowship" -- "hey let's drink together", but never on true friendship.
There will always be some distance as we don't want to blur the lines of our faith, but that doesn't mean "you can't have non-Jews as friends." The Talmud describes the warm relationship between Rabbi Yehuda HaNasi and a Roman official named Antoninus, for example.