Gemara

(Beraisa): Sinners of Yisrael may bring Korbanos, in order that they repent;

We do not accept Korbanos from a Mumar, one who is Menasech (offers libations to idolatry), or one who desecrates Shabbos in public.

39a (Reish Lakish): If an animal was slaughtered with intent to throw its blood for idolatry, it is permitted.

Bava Kama 113b (Rav Chama bar Gurya): One may keep an Aveidah of a Nochri. We learn from "l'Chol Aveidas Achicha." You return an Aveidah of your brother, but not an Aveidah of a Nochri.

Question: Perhaps that only exempts one from taking the object to return it. What is the source that one who took it may keep it?

Answer (Ravina): "And you found it" connotes that he already picked it up (and still, you must return only to your brother).

(Beraisa - R. Pinchas ben Ya'ir): If Chilul Hash-m (desecration of Hash-m's name) will result, even keeping a Nochri's Aveidah is forbidden.

Sanhedrin 76b (Rav Yehudah): "Lema'an Sefos ha'Ravah Es ha'Tzeme'ah" refers to one who returns an Aveidah to a Nochri.

Avodah Zarah 26a (R. Avahu - Beraisa): We throw into a pit Minim (priests of idolatry)... and Mumrim (people who habitually transgress a certain Mitzvah).

Objection (R. Yochanan - Beraisa): "L'Chol Aveidas Achicha" includes the Aveidah (lost object) of a Mumar.

We must return his property. All the more so we do not kill him!

Question: Perhaps we must return the Aveidah of a Mumar who eats Neveilos l'Te'avon (to fulfill his appetite), and the Beraisa teaches that we kill a Mumar who eats Neveilos Lehach'is (to anger Hash-m)!

Answer: R. Yochanan holds that a Mumar Lehach'is is considered a Min (which was already mentioned in R. Avahu's Beraisa).

Rishonim

Rif and Rosh (Avodah Zarah 8a and 2:7): We learn from our Gemara that all agree that "Aveidas Achicha" does not apply to one who eats Neveilos Lehach'is.

Ran (DH lta): If one eats Neveilah when he has no other meat, this is l'Te'avon. If he has also Heter and he eats Isur, this is Lehach'is.

Rosh: If a Mumar is Kofer b'Ikar (denies basic tenets of Torah), one may lend to him on Ribis (interest). One may not take Ribis from a Yisrael, due to "v'Chai Achicha Imach." There is no Mitzvah to help a Mumar to idolatry live. However, one may not pay Ribis to a Mumar. Even though he sinned, he is a Yisrael. The Isurim of Lifnei Iver and paying Ribis apply.

Rambam (Hilchos Gezeilah 11:2): It is a Mitzvah to return a Yisrael's Aveidah, even if he is a Rasha who eats Neveilos l'Te'avon and similar matters. If he eats Neveilah Lehach'is, he is a Min. One may not return an Aveidah to a Yisrael Min, Apikoros, idolater, or one who is Mechalel Shabbos in public, just like one may not return to a Nochri.

Rambam (3): An Aveidah of a Nochri is permitted (one may keep it), for it says "Aveidas Achicha." One who returns it transgresses, for he supports Resha'im. If he returns it to be Mekadesh Hash-m, so people will glorify Yisrael that they are trustworthy, this is praiseworthy. If Chilul Hash-m will result from keeping it, one must return it. In every place we bring in their Kelim to save them from thieves, like Yisre'elim's Kelim, due to Darchei Shalom.

Rambam (Hilchos Isurei Bi'ah 13:17): If a convert reverted to serve idolatry, he is like a Yisrael Mumar. If he was Mekadesh a woman, she is Mekudeshes. It is a Mitzvah to return his Aveidah, since he became a Yisrael.

Magid Mishneh: This is like the Rambam says in Hilchos Gezeilah. We return to a Mumar who eats Neveilos l'Te'avon.

Rebuttal (Kesef Mishneh): L'Te'avon does not apply to idolatry. There is no enjoyment from it! Also, an idolater or Mechalel Shabbos for any reason is like a Nochri! Rather, the Rambam teaches that one must return a convert's Aveidah; this law does not discuss an idolater.

Defense (Bach YD 268 DH Echad): Yisrael lusted for Arayos and theft. They served idolatry so that Beis Din would not rebuke them for other Aveiros. The Magid Mishneh explains that the Rambam obligates returning a convert's Aveidah if he is a Mumar to eat Neveilos l'Te'avon.

Tosfos (Chulin 14a DH ha'Shochet): The Gemara suggests that one may eat an animal slaughtered on Shabbos even if this was b'Mezid, according to R. Meir. R. Meir holds that a Mumar to one matter is a Mumar to the entire Torah! Perhaps that is when he does so publicly; here we discuss one who slaughtered covertly. Also, we can say that one who sinned once is not a Mumar. Chulin 39a supports this. Reish Lakish permits (eating) an animal slaughtered with intent to throw the blood to idolatry, even though the one who slaughtered is Chayav Misah. We ask from a Beraisa that forbids, and answer that a Mumar's Shechitah is forbidden. This is a good answer if one who sinned once is not a Mumar.

Mordechai (Chulin 573): If one served idolatry or was Mechalel Shabbos once, and he does not freely do so repeatedly in public, he is not a Mumar to forbid what he slaughtered. We learn from Reish Lakish. A Beraisa refuted Rav Anan, who permitted what a Mumar slaughtered. We do not refute Reish Lakish from the Beraisa (for one who sinned once is not a Mumar). If one cooked or slaughtered on Shabbos, others may eat from it.

Poskim

Shulchan Aruch (CM 266:1): An Aveidah of a Nochri is permitted, for it says "Aveidas Achicha." One who returns it transgresses, for he supports transgressors.

Beis Yosef (DH Aveidas): Clearly, all Nochrim are the same, whether or not they serve idolatry, for they are not Achicha. The Tur wrote 'idolaters'; this is not precise. Perhaps this is because he lived in Eretz Edom. There, Minim used to malign Yisrael in the eyes of the kingdom due to laws like this, and Chachamim answered that this was only in the days of the Gemara when Nochrim served idolatry and did not admit to the Creator, but nowadays the Nochrim are not idolaters. Alternatively, the Tur said 'idolater' to include Yisrael idolaters.

Beis Yosef (DH u'Mah she'Chosav v'Lo): Rashi explains that one who returns an Aveidah to Nochrim equates them to Yisre'elim, and shows that he does not consider Hashavas Aveidah to be His Creator's Mitzvah, for he does so even to Nochrim, for whom there is no Mitzvah.

Pischei Teshuvah (1, citing R. Akiva Eiger Bava Metzia 22b): If Reuven was guarding a Nochri's item and he lost it, and he will be liable if he cannot return it, it is a Mitzvah to return it to save Reuven from a loss. This is even like Chachamim of R. Shimon, who say that Davar ha'Gorem l'Mamon (something that can save from a loss) is unlike money. Rav Yisrael disagrees, for it is not called Aveidas Achicha. This requires investigation.

Shulchan Aruch (ibid): If he returns it to be Mekadesh Hash-m, so people will glorify Yisrael that they are trustworthy, this is praiseworthy. If Chilul Hash-m will result from keeping it, one must return it.

Beis Yosef (DH u'Mah she'Chosav b'Shem): The Yerushalmi says that it is praiseworthy to return it to be Mekadesh Hash-m. The Chiyuv to return it to avoid Chilul Hash-m is even if he did not yet pick it up, e.g. in a place where there are mostly Yisre'elim, and if he does not return it, the Nochri will think that a Yisrael stole it.

Shulchan Aruch (ibid): In every place we bring in their Kelim to save them from thieves, like Yisre'elim's Kelim, due to Darchei Shalom.

Beis Yosef (DH u'Mah she'Chosav uv'Kol): This is from the Yerushalmi.

SMA (4): The Tur holds that the Torah forbids only a Nochri's Aveidah that fell from the owner without his knowledge, but not when one knowingly leaves his Kelim is his Chatzer, and a Yisrael knows that thieves will come and take them. Since it is not considered an Aveidah, one may take them to a guarded place, due to Darchei Shalom. It is permitted everywhere, i.e. even where Chilul or Kidush Hash-m does not apply. One need not bring in a Yisrael's Kelim, i.e. if they are guarded in their place.

Drishah (2): It seems that we bring in Nochrim' Kelim not only along with Yisrael Kelim. Rather, since we do so for Yisre'elim, there will be enmity if we do not do so also for Nochrim.

Shulchan Aruch (2): One must return a Yisrael's Aveidah, even if the owner is a Rasha who eats Neveilos l'Te'avon. If he eats Lehach'is, he is a Yisrael Apikoros. One may not return an Aveidah to an Apikoros, idolater, or one who is Mechalel Shabbos in public, just like one may not return to a Nochri.

Beis Yosef (DH u'Mah she'Chosav v'Ovdei): In Chulin (5a) we say that one who is Mechalel Shabbos in public is like a Nochri.

SMA (6): 'In public' refers also to idolatry. If he served only once, he is a Mumar only if it was in public. Tosfos and the Mordechai say so.