1) A KOHEN WHO IS TAMEI WITH "TUM'AS HA'TEHOM"
QUESTION: The Mishnah (80b) states that if a person discovers that he was Tamei with Tum'as ha'Tehom when he offered his Korban Pesach, his Korban nevertheless remains valid and he does not have to bring another Korban on Pesach Sheni, because the Tzitz is Meratzeh for Tum'as ha'Tehom. The Gemara asks whether the Tzitz is also Meratzeh for a Kohen who discovers that he was Tamei with Tum'as ha'Tehom when he offered the Korban Tamid.
RASHI (DH b'Tamid) explains that the Gemara's question applies only when the Kohen finds out about his Tum'ah before he performs the Zerikah of the blood of the Korban Tamid. The Gemara's question applies according to the opinion (81b) that one may perform the Avodah even l'Chatchilah with Tum'as ha'Tehom when he finds out about it before the Zerikah, and according to the opinion that "Tum'ah Dechuyah b'Tzibur" (as opposed to "Hutrah").
Why does Rashi explain that the Gemara's question applies specifically when the Kohen finds out about his Tum'ah before the Zerikah? Why does Rashi not explain that the Gemara's question about the status of the Korban applies when the Kohen finds out about his Tum'ah after he offers the Korban?
ANSWER: TOSFOS in Zevachim (23a, DH Lo) writes that if the Kohen first offered the Korban and then discovered that he was Tamei, the Korban certainly is valid, because that is the normal application of the rule that Tum'ah is disregarded for a Tzibur. What, then, is the Gemara's question? The Korban Tamid certainly is valid when it was offered by a Kohen who was Tamei with Tum'as ha'Tehom!
Rashi and Tosfos therefore explain that the Gemara's question of Tum'as ha'Tehom refers only to the opinion that "Tum'ah Dechuyah b'Tzibur." That opinion maintains that Tum'ah must be avoided whenever possible, and only when there is no other way to offer the Korban may it be offered b'Tum'ah. Accordingly, the Gemara asks about a case in which the Tum'as ha'Tehom is discovered before the Zerikah is performed, at a time when the Korban can still be offered b'Taharah.
2) WHEN DOES "TUM'AS HA'TEHOM" APPLY
OPINIONS: The Gemara cites a Beraisa which states that if a person walks along a path that was later found to have Tum'as ha'Tehom (a corpse was found spread lengthwise from one side of the path to the other, which no one previously saw or knew about), the person is considered Tahor with regard to the Korban Pesach. With regard to Terumah, however, he is considered Tamei, because we assume that he walked over the corpse and became Tamei.
The law of Tum'as ha'Tehom, Tum'ah that was unknown or hidden, is a Halachah l'Moshe mi'Sinai. Under what conditions does the law of Tum'as ha'Tehom apply?
(a) RASHI (DH she'Ein) explains that the Halachah of Tum'as ha'Tehom applies post facto, after the person slaughters his Korban Pesach. When he discovers after he slaughtered the Korban Pesach that he was Tamei with Tum'as ha'Tehom, even though he was definitely Tamei at the time of the Shechitah, the rest of the Avodah of the Korban (i.e. the Zerikah) may be done on his behalf. The Halachah l'Moshe mi'Sinai of Tum'as ha'Tehom teaches that he fulfills the Mitzvah to offer the Korban Pesach even though he was Tamei, and he is not required to bring another Korban on Pesach Sheni.
The law of Tum'as ha'Tehom is a Halachah l'Moshe mi'Sinai and is not based on practical considerations. Nevertheless, the underlying theory of the law of Tum'as ha'Tehom, and why the Korban Pesach of a person who was Tamei with Tum'as ha'Tehom is valid, is that the person was Karov l'Ones -- the circumstances under which he slaughtered the Pesach while he was Tamei were almost completely beyond his control. It was impossible for him, or anyone, to know that he was Tamei at the time he slaughtered the Korban. Therefore, the Korban is acceptable even though he was Tamei. (This leniency with regard to Tum'ah applies only to the Korban Pesach, because the consequences of not bringing the Korban are so severe.)
(b) The RAMBAM (Hilchos Korban Pesach 6:11) explains that the leniency of Tum'as ha'Tehom applies to a person before he slaughters his Korban Pesach, and is limited to a case of a doubt of Tum'ah. Before the person slaughters the Korban, he discovers that he is Safek Tamei with Tum'as ha'Tehom. The Rambam maintains that his Korban is valid only when he is not certain that he was Tamei. The law of Tum'as ha'Tehom applies only where the Tum'ah was unknown heretofore, and even now that the Tum'ah was discovered, it is not certain that the person became Tamei. Although the Beraisa says that a corpse was stretched across the width of the path, it was spread in such a way that it is possible that the person did not walk over it and become Tamei.
According to the Rambam's understanding, the theory behind the Halachah l'Moshe mi'Sinai of Tum'as ha'Tehom is that since it is so important to bring the Korban Pesach (because of the severity of its punishment, Kares), the Torah says that if there is even a small possibility that the person is not Tamei, he is allowed to bring the Korban. For Terumah, though, the doubt is not great enough to permit him to eat Terumah, because refraining from eating Terumah is not at all as severe as failure to bring the Korban Pesach.
The CHAFETZ CHAIM (in ZEVACH TODAH) points out that there exists a stringency and a leniency according to each opinion. According to Rashi, if one discovers that he is Tamei (or even Safek Tamei) with Tum'as ha'Tehom before the Shechitah, he may not offer the Korban (a stringency), while according to the Rambam he may offer it. According to the Rambam, if one discovers that he is certainly Tamei with Tum'as ha'Tehom, even after the Shechitah, the Korban that he offered is not valid and he must bring another one (a stringency), while according to Rashi the Korban is valid.