Believing that one is a dragon does not, in itself, create incompatibilities with any religion -- there is no dogma of draconity that states what you need to believe. There are, of course, conflicts with some faiths, but these are caused by the dogma of the religion in question, most often when the religion labels dragons as inherently evil.

To provide a specific example: Christian churches that call for a literal interpretation of the Bible traditionally have had problems with draconity. (See following questions.)