Gemara

(Mishnah): If the son of a Kohen became mixed with his Shifchah's son, both of them may eat Terumah. They receive one share at the granaries;

They may not become Tamei Mes. They cannot marry Kosher or Pasul women.

If they grew up and freed each other, the following laws apply to each:

He marries a woman fit for a Kohen;

He may not become Tamei. If he did, he is not lashed;

He may not eat Terumah. If he ate, he does not pay the principal, nor an added fifth;

He does not receive a share of Terumah at the granaries. He sells Terumah that he separates from his own produce, and keeps the money;

He does not get a share of Kodshim of the Mikdash. If he comes to offer a Korban, we do not seize it from him;

He is exempt from giving Matanos (the foreleg, jaw and stomach) when he slaughters a Chulin animal;

His firstborn animals graze until they get a blemish (and then he eats them);

We put on him the stringencies of a Kohen and of a Yisrael.

Kidushin 75a (Beraisa): If a boy above nine years old who is forbidden to any Yisraelis (e.g. he is an Amoni or Mamzer) had Bi'ah with a Yisraelis, he disqualified her (from Kehunah);

R. Yosi says, he disqualifies her only if his children are disqualified;

R. Shimon ben Gamliel says, if a man's daughter is permitted to a Kohen, his widow is also. If his daughter is forbidden, so is his widow.

(Rav Chisda): All agree that the widow of an Isah (a family in which a doubtful Chalal was mixed in) is forbidden to Kehunah.

R. Shimon ben Gamliel is the most lenient of the above Tana'im. He said '...if his daughter is permitted, his widow is also', to exclude the widow of an Isah, who is forbidden to Kehunah.

Rav Chisda teaches that we do not hold like the following Tana'im.

(Mishnah): R. Yehoshua and R. Yehudah ben Beseira testified that the widow of an Isah is permitted to Kehunah.

They permit her because it is a double Safek. (Perhaps she did not marry the doubtful Chalal. And even if she did, perhaps he was really Kosher.)

Rishonim

Rambam (Hilchos Isurei Bi'ah 17:7): If a Kohen married a Safek Chalutzah, he may keep her. She and her children are Kesherim. Chachamim did not decree about a Safek Chalutzah.

Rambam (19:9): If a Kohen had Bi'ah with a woman who is a Safek Zonah, convert, freed slave, or divorcee, or if a Kohen Gadol had Bi'ah with a Safek widow, she is a Safek Chalah and her son is a Safek Chalal.

Magid Mishneh: We learn that there is a Safek Chalal from a widow of an Isah.

Note: The Magid Mishneh cites Kesuvos (14b). Perhaps this is a Chidush, for the Torah permits a Safek Mamzer, and the Rambam learns from this that the Torah permits (most) Sefekos. The next Halachah shows that the Rambam holds that a Safek Chalal is not just mid'Rabanan. Tosfos (brought below) learns from Kidushin 75a that we forbid the daughter of a Safek Chalal.

Rambam (10): There are Chalalim mid'Oraisa, mid'Rabanan and Safek Chalalim. A Safek Chalal has the stringencies of Kohanim and of Yisraelim. He may not eat Terumah or become Tamei Mes. He marries a woman Kosher for Kohanim. If he ate, became Tamei Mes or one married a divorcee he is lashed mid'Rabanan.

Magid Mishneh: This is no less than an Isur mid'Rabanan, for which one is lashed mid'Rabanan.

Rambam (20:17): Leah married Levi within three months of leaving David. If we do not know whether her child (Moshe) is a nine-month baby from David or a seven-month baby from Levi, if exactly one of Levi and David is a Kohen, Moshe is a Safek Kohen. Similarly, if a Kohen's son became mixed with a Yisrael's son, both boys are Safek Kohanim. They have the stringencies of Kohanim and of Yisraelim. They marry women Kosher for Kohanim. They may not become Tamei Mes or eat Terumah. If one married a divorcee he must divorce her, but he is not lashed.

Tosfos (Kesuvos 14a DH Almanas): It seems that even the opinion that permits the widow of an Isah forbids the daughter. Even though both are Sefek-Sefekos, it is a stringency of lineage to forbid the daughter. We do not forbid the mother, because she has a Chezkas Kashrus.

Poskim

Shulchan Aruch (EH 3:8): If Leah married Levi within three months of leaving David and we do not know if her child Moshe is from David or from Levi, if only one of Levi and David is a Kohen, Moshe is a Safek Kohen. Similarly, if a Kohen's son became mixed with a Yisrael's son, both boys are Safek Kohanim. They have the stringencies of Kohanim and of Yisraelim. They marry women Kosher for Kohanim. They may not become Tamei Mes or eat Terumah. If one married a divorcee he must divorce her, but he is not lashed.

Gra (28): The Gemara discussed when one became Tamei. The same applies to other Isurim.

Shulchan Aruch (7:20): If a Kohen had Bi'ah with a Safek Zonah (Rema - or a Safek Chalutzah), she is a Safek Chalalah and her son is a Safek Chalal.

Rebuttal (of Rema - Gra 65): B'Di'eved, Chachamim permitted a Safek Chalutzah to a Kohen. Her son is like b'Di'eved, he is also permitted! This is why the Rambam omitted this case.

Bach (DH Kohen): The Rambam (17:7) is explicitly Machshir the son of a Safek Chalutzah. We learn from Yevamos 23b. Why does the Tur disqualify? The Tur must hold that we Machshir only a child through marriage, but not a child through Zenus.

Rebuttal (Mishneh l'Melech 17:7): This distinction is unreasonable. The Shulchan Aruch does not disqualify the child; I do not know why the Rema disqualifies like the Tur. Perhaps the correct text of the Tur disqualifies the son of a Safek Chalahah

Shulchan Aruch (ibid.): He has the stringencies of Kohanim and of Yisraelim. He may not eat Terumah or become Tamei Mes. He marries a woman Kosher for Kohanim. If he ate (Terumah) or became Tamei Mes married a divorcee, he is lashed mid'Rabanan. The same applies to a Chalal mid'Rabanan.

Prishah (37): Mid'Oraisa, a Chalal mid'Rabanan is a Kosher Kohen. He should be lashed mid'Oraisa! It seems that Chachamim made him totally like a Safek Chalal (and lash only mid'Rabanan), lest people say that he is a proper Kohen and feed to him Terumah.

Rebuttal (Bach ibid., Chelkas Mechokek 28, Taz 15, and Beis Shmuel 40): He is lashed mid'Oraisa. 'The same applies' refers to the Isur l'Chatchilah, not to lashes.