Christianity, in general, demands a belief in Jesus Christ as one''s savior and in the unity and supremacy of God. Draconity is not incompatible with either of these concepts, and so being a Christian dragon is certainly possible. "Praise the Lord from the earth, ye dragons, and all deeps." (Psalm 148:7)
It is far simpler to be a dragon, to be certain, in a liberal church. In more conservative organizations, which tend to insist on a literal interpretation of the Bible, draconity can be incompatible with the teachings of the institution (see next question).
Christian dragons, in my experience, tend to have a very colorful and living mythology that unites their conception of dragon with their conception of God. One friend of mine, for example, identifies dragons as among God''s angels.