There is an idea mentioned by the Rambam, but not in the Mishneh Torah as I recall, that it is permissible to teach Wrtitten Torah to Christians because they accept it as Divine and don't study it in a strictly academic/detached way. Where does he, and does he in fact, rule this way? How would this apply to teaching Oral Torah to someone such as a b'nei Noach who accepts it?
The source you are looking for is the Rambam's responsa (§149), where the Rambam writes:
When the Rambam writes, "וכאשר יד ישראל תקיפה עליהם מונעים אותו מתלמוד תורה עד שיתגייר", I'm not sure if the Rambam meant that we don't teach non-Jews Torah until the conversion process is complete, or if he meant that we don't teach them Torah until they come seeking to convert. In any case, it appears likely from this responsum, and from the the Rambam's ruling in Hil. M'lachim 10:9, that even a ben Noach who accepts the divinity of the Torah (see Hil. M'lachim 8:11) may not study Torah other than as pertains to him (however, the scope of what Torah pertains to a non-Jew may be quite broad - see this answer).
If so, perhaps we could reconcile this with the Rambam's remarks about Christians by saying that, absent a need to correct someone's aberrant understanding of the Torah, we don't teach Torah (and, even then, teaching them is conditional on their prior acceptance of our "textus receptus" of the Old Testament).
The motivation for allowing correcting Christians' mistaken understanding of the Torah may be related to the teaching in Avos (2:14) to "know how to respond to a heretic". The Rambam (ad loc.) comments that "you must learn what you will need to know to respond to non-Jewish heretics, and dispute them and reply to them if they pose a challenge to you".
1 According to halacha, non-Jews are obligated to establish secular courts of law. These courts are theoretically empowered by halacha to punish transgressors of the seven Noahide laws under certain circumstances.
2 This can also be translated as "stranger", meaning non-Jew.
It is not only permissible to teach written Torah to the Gentiles but also highly recommendable. Prophet Isaiah himself says that in the last days, when the mountain of the Lord shall be established in the top of the mountains, many people shall go and say, "Come, let us go up to the mountain of the Lord, to the House of the G-d of Yakov, and He will teach us of His ways, and we will walk in His paths; for out of Zion shall go forth the Law, and the Word of the Lord from Jerusalem." (Isa. 2:2,3) "And in those days it shall come to pass that the nations shall take hold of the garment of him who is a Jew and beg of him to let them join Israel, because they have finally acknowledged that HaShem is with us." (Zech. 8:23) I know we have stopped proselytizing Gentiles, because of Antisemitism. But if we think it through, the best way to fight Antisemitism is by making a Jew out of the anti-Semite.
A few years ago, a family of five Christians from the Seventh-Day Adventist Church knocked at my door and, very politely, asked me to be allowed to speak a word to me about Jesus. I invited them in. It was on a Shabbat afternoon. After about two months, I realized a change in their attitude. They were listening more and speaking very little, and more in terms of questions. They ne'elemu (disappeared) for about eitht months when I got an invitation for their ceremony of conversion. And Orthodox style conversion, according to Halacha, mind you. And I was presented as the reason why they had decided to join Judaism. The two daughters of the family got married with two nice Israeli youngmen, while I was chosen to be the best man. Today, each one has two born Jewish children. It means I have contributed with nine very happy souls to Judaism. Baruch HaShem.
The Maharatz Chiyos to Sotah 35b writes that a Jew can teach Tanach to a gentile. The Netziv (Meshiv Davar, Yoreh Deah 77) and R. Yehudah Assad (Yehudah Ya’aleh, Orah Hayim 4) agree. Also the Rambam in Responsa Pe’er Ha-Dor 50. The Meor Veshemesh (Parshas Chukas) writes that it is permitted to teach the Written Law to an idolater. This is like Moshe who wrote the Torah in seventy languages. The prohibition of teaching Torah to Christians applies only to the Oral Law. Also the Netziv again in his Pirush Al Hatorah in Vayikra 18:5 says explicitly that a non-Jew can learn Torah shebiktav. Also The Divrei Chaim (Chanukah) says that the Torah was written on the stones and the nations of the world copied it over. The Medrash says that Hashem allowed them to study the Written Law. For more See here: http://dafnotes.blogspot.com/2007/06/teaching-torah-to-gentile-planning-on.html And just to be Jewish you Know that the gemara says a non-Jew who learns torah is like the Kohen Gadol? (A"Z 3a) ?