I think I may have resolved the issue of the surprising apparent ignorance of the Ran regarding Muslim belief and practice: In another opinion, the Ran holds that Islam is not idolatry and the citation given by Kasha is not by the Ran himself but by a later (presumably less knowledgeable) author. Details are in Prof. Marc Shapiro’s article, “Islam and the halakhah” in *Judaism”, Summer 1993. It is available online.
A quote from the article:
‘What about a Jew entering a mosque? According to the halakhah, a Jew is forbidden to enter a house of idolatry and, therefore, almost all halakhic authorities forbid one from entering a church. Since Islam is not idolatrous, there should be no problem for a Jew to enter a mosque and, although no early halakhists seem to discuss this question, the prevailing opinion of recent halakhic authorities is to be lenient. Similarly, one nineteenth century Ashkenazic authority permitted the conversion of a mosque to a synagogue without the “nullification” of idolatry that is required when converting a church into a synagogue, while another authority even permitted a Jew to assist in the building of a mosque.’
The actual article includes citations. It does not, however, deal with the original question regarding Korans. Hopefully no Jew will consider the silence on the matter to indicate permission; if we allow Koran burnings you can be assured that sifrei torah will be next.