I know that some mitzvot like keeping shabbat, putting tefillin are forbidden for a gentile to do because they are a sign of the covenant between Israel and HaShem.
What are the full list of the forbidden mitzvot ?
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The Raadbaz was choichech and decided to be stringent in regards to tefillin, mezuzot, and scrolls(Raadbaz on Hilchot Melachim 10:10). This is in regards to bnei Noach and not goyim(Hilchot Melachim 10:10). Rambam states that bnei Noach can do any of the mitzvot as long as they do it properly(Hilchot Melachim 10:10). Goyim, which are idol-worshipping non-Jews(Hilchot Machalot Assurot 11:8), should not be doing anything but being involved in the seven laws of the children of Noah(Hilchot Melachim 10:9). A person should be pure before engaging in holiness.
It's my personal opinion that the Raadbaz was choichech(he deliberated) on the matter because there are two main types of bnei Noach. Bnei Noach that do things for their own sake and bnei Noach that do things for the sake of Torah(Hilchot Melachim 8:10-12). These days, because the Yovel year is not practiced a Jewish beit din cannot bring the fear to the ben Noach to properly determine his kavanah(intention). Such a ben Noach accepted would be a ger toshav and then part of klal Yisrael. The other type of ben Noach would most likely not come before the beit din. This is why I think the Raadbaz was being stringent.
All goyim by default, are idol-worshippers until said otherwise(Gittin 45b, Hullin 13a, Hilchot Maachalot Assurot 11:8, R"ITVA on Makos 9a, Chasam Sofer on Sanhedrin). However, that does not mean they're always idol-worshippers. They could explicitly state themselves as not idol-worshippers. Once they do that, they come out of the geder/category of goy and into ben Noach.