Can a non-Jew recite the Shema and other Jewish prayers? Is there any one which is off limits to non-Jews? Thanks.
The Sh'ma is a passage from the torah, not specifically a prayer, and in fact when we say it we are technically studying torah (citation needed, sorry). Reading, silently or aloud, passages from Tanakh isn't restricted to Jews, as evidenced by the lack of outcry from Jews when Christians quote our texts all the time. (There may be outcry about their interpretations, of course, but not that they read a passage from D'varim aloud.)
As for prayers, some use the phrasing "praised is God... who has sanctified us with the commandment to X". These are inappropriate for a non-Jew to say because God hasn't given the non-Jew that commandment. Other prayers are not of this form and I believe there is more flexibility there.
However, it is forbidden for a non-Jew to say either type of prayer on behalf of the community; the community cannot fulfill its obligation to bless through someone who is not obligated. (This is why, when there is a bar mitzvah in a family with a convert or an intermarriage, the non-Jewish relatives are not allowed to recite the torah blessings.)
It's worth noting that a person in the process of converting will begin to say that class of prayers for himself well in advance of conversion, but not on behalf of others.