I read on Mi Yodeya, and elsewhere on the Internet, that the Gemara (Chagigah 13a) says that Jews can't teach Torah to gentiles, and that according to Tosefot (Chagigah 13a sv. ein) this is because the Torah belongs to the Jewish people as part of your national uniqueness. Studying Torah is something that shouldn't be stolen by gentiles so as to blur the lines of and lessen the distinctness of the calling you've been given in the world and in the kingdom of God. The Rambam held (in Mishneh Torah) that gentiles can do many Jewish practices out of curiosity or as an optional mitzvah, but absolutely not Torah study or keeping Shabbos.
I'm not sure what the connection is here between oral Torah and written Torah. I know that some rabbis have made distinctions between Torah that is relevant to all humanity and that which is only instructive for Jewish practice (and forbidden to gentiles). I also assume that there are some levels and methods of passed on tradition that may have been held more closely to the identity of Jewish Torah learning than other widely known things.
This brings me to two questions: one is personal, the other is in the context of Jewish community. The situation is made unclear by a number of things. Many gentiles who are interested in Judaism have grown up with a form of Tanach and of actual faith in it (though this was already an issue for the Tosafists), and so feel both entitled and connected to it. Specific examples might also be helpful for answering confusion about the oral Law. Second, Jews have wanted to share the light of who God is, and also to avoid persecution and unpleasant relationships, by making a space for gentiles to hear Torah and Judaism. Third, there are gentiles who want either to convert or to be connected with the Jewish community through belief (these are two quite separate cases in a question like this), and so deliberately spend time with and learn from (and with) the community.
So personally, if I make the choice to honour the opinions of the leaders of Judaism about what is appropriate for gentiles to do and about what is most honouring to the program that God has set out for Israel... how do I know what to read, both out of devotion (not to practice Judaism but to recognise the light it sheds on the human relationship with God) and out of curiosity (academic or in understanding the community I'm in relationship with)? It seems basic, but at a foundational level, is it appropriate for gentiles to read (all of) Tanach? If so, how about reading books about Jewish law, or the Talmud itself; issues pertaining to gentiles directly, or laws pertaining only to Jews that are still meaningful to read? Listening to yeshiva-type classes about these things online, if I chose to? How about reading Kabbalistic works, or Jewish devotional literature? Listening to people talking with their kids about Torah on Shabbos, or coming to shiurim (which I've been allowed to do)? I recognise that there are many opinions that would allow some or all of these things, but I haven't learnt about how I'm meant to deal with a multiplicity of authorities that allowed and prohibited the one thing. Surely I can't just glean and put together opinions to rely on that are most convenient to me, such as the idea that gentiles are fine reading what we want to out of curiosity and that we can be in a mixed group of Jews and gentiles learning, or that we can be taught certain material. If I take those views, then what about the original concern that something might be diminished from the Jewish testimony by allowing gentiles in freely to its circles? What does God want people like me to be doing, and on what basis does the Jewish community give me a unified decision about what is considered appropriate by you with the things you guard? This is important to me both in my relationship with God and in my decisions regarding study at uni. (It's already complicated with knowing how to engage with secular and other religions' material in a healthy way, both for enjoyment of and identification with those things and with my own culture, and for studying them as literature.)
The other question is not about what I should do, but rather what Jewish leaders are comfortable with people doing in your own community. Some of the situations written about in the paragraph above are relevant in this way. It doesn't matter what is convenient or polite or 'done everywhere', it matters what is really right and good in the bigger context, with precedents appropriately translated across. I have asked some rabbis what they think about this as well, and I'll see what they think, but I wouldn't mind hearing as many angles, thoughts, and sources about this as could be found through Mi Yodeya.
This is related to the question I posted yesterday. In both of these, if I know what is open to me and what is closed, in a way that best serves and guards the Jewish relationship with Hashem, then I can find breadth in relationship with Him and with your community within that space that is left.
It comes down both to what is instructed in a binding way, and to what is most honouring to God (and to people in Judaism) in His plan and His ways. This is difficult for me because my foundations have been shaken a lot this year in being challenged by Judaism not to hold to Christianity, so on the one hand I'm walking near to God's heart and wanting to honour Him according to what is revealed in the sphere of Judaism as well as the things I sense naturally; on the other hand, I wonder how much of this He really revealed, and how much is human traditions that I'm merely involved with and abiding by because of habit, community, stability, and things like that. That's hard to deal with in the choice to live in faith and faithfulness. Anyway. It's an important conversation, answers are welcome :)