Gemara

(Mishnah): One who is suspected about Shemitah is not suspected about Ma'aser (Sheni);

One who is suspected about Ma'aser is not suspected about Shemitah;

One who is suspected about either of these is suspected about Taharos (not to eat Chulin b'Taharah);

Someone can be suspected about Taharos, but not about either of these.

Question: Why is someone suspected about Shemitah not suspected about Ma'aser?

Answer: Shemitah need not be eaten within the wall of Yerushalayim, but Ma'aser must. Perhaps he considers Ma'aser to be more stringent.

Question: Why is someone suspected about Ma'aser not suspected about Shemitah?

Answer: Ma'aser can be redeemed. Shemitah cannot be redeemed once it becomes forbidden (e.g. after Bi'ur). Perhaps he considers Shemitah to be more stringent.

(Mishnah): One who is suspected about either of these (is suspected about Taharos).

Since he is suspected to transgress a mid'Oraisa law, all the more so he is suspected about mid'Rabanan laws.

(Mishnah): One can be suspected about Taharos (but not about either of these).

Even though he is suspected about a mid'Rabanan law, but he is not suspected about mid'Oraisa laws.

Version #1 (Rabah bar bar Chanah): Our Stam Mishnah is like R. Akiva, but Chachamim say that one who is suspected about Shemitah is suspected about Ma'aser.

These Chachamim hold like R. Yehudah. In his area, Shemitah was severe to people;

There, someone once called someone "convert, son of a convert" (to insult him). The latter responded "I don't eat Shemitah produce like you do!"

Version #2 (Rabah bar bar Chanah): Our Stam Mishnah is like R. Akiva, but Chachamim say that one who is suspected about Ma'aser is suspected about Shemitah.

These Chachamim hold like R. Meir, who says that one who is suspected about one matter is suspected about every Mitzvah.

R. Yonah or R. Yirmeyah explained like Version #1, and the other explained like Version #2.

Chulin 4b (Beraisa): Anyone may slaughter, even a Kusi, Arel (uncircumcised man), and even a Mumar.

The Arel rejects the Mitzvah of circumcision. The Tana holds that one who rejects one Mitzvah is not like one who rejects the entire Torah.

Question: The Seifa says, 'even a Yisrael Mumar.' What is the case?

If he is a Mumar to a Mitzvah other than Nevelah, this is the same as Mumar to circumcision, which was already taught!

Answer: Rather, he is a Mumar to eat Nevelah, and it says that he may slaughter!

Rejection: No. Really, a Mumar to eat Nevelah cannot be trusted, for he considers it to be permitted! Rather, the Seifa discusses a Mumar to idolatry. It supports Rav Anan;

(Rav Anan): If a Mumar to idolatry slaughters, the animal is Kosher.

5a - Rejection: No. Really, a Mumar to idolatry may not slaughter. Anyone who admits to idolatry is like one who denies the whole Torah;

Rather, the Seifa discusses a Mumar to eat Nevelah (for pleasure), like Rava taught (4a. One checks a knife, and the Mumar slaughters with it.)

(Beraisa): We do not accept Korbanos from a Mumar, one who is Menasech (offers wine libations to idolatry), or one who publicly desecrates Shabbos.

This means that a Mumar who is Menasech or publicly desecrates Shabbos (may not bring Korbanos).

Rishonim

Rashba (Teshuvah 170 attributed to the Ramban, brought in Beis Yosef YD 119 DH Kasuv bi'Teshuvos): One suspected of a matter is not suspected of something more severe, and not even of something less severe. A Mumar for Orlos (circumcision) is not a Mumar for Shechitah. We say that a Mumar to transgress one matter is a Mumar to the entire Torah only if he is a Mumar for Yayin Nesech or to be Mechalel Shabbos in public (Chulin 5a). However, regarding what he is suspected about, he is like a total Mumar. He is not believed even with an oath, for he is suspected to transgress the oath from Sinai about it. If you would not say so, a Mumar to eat Nevelah could swear that he will not eat Nevelah, and we would not need to check (the knife he used for Shechitah) every time. Why do we forbid the animal if it was not checked? He can swear that he slaughtered properly with a good knife! Amei ha'Aretz are not believed about Ma'aser. Why can't they swear that they tithed? Further, even if one will not eat (what is not tithed) or feed it to others, we do not accept him until he accepts Chaverus in front of three. It does not suffice to swear in front of two!

Rivash (12, cited in Beis Yosef ibid.): If so, if Anusim (Yisre'elim who openly became Notzrim due to coercion, but kept the Mitzvos in private) are suspected about Stam wine (of Nochrim), and even about whether a Nochri touched their wine, then they are not believed even through an oath. However, they are believed about others', for one does not sin if he does not benefit from it.

Beis Yosef (DH v'Al): The Ramban (really, Rashba) said that one suspected of a matter is not suspected of something more severe, and not even of something less severe. This is difficult, for one who is suspected about Ma'aser or Shemitah is suspected about Taharos. The Gemara says that this is because one who is suspected about Torah laws is suspected about mid'Rabanan laws. Perhaps he holds that we say so only from Torah to mid'Rabanan laws, but not if both are mid'Oraisa or both are mid'Rabanan, and one is more lenient. However, the Gemara explained that one who is suspected about Shemitah is not suspected about Ma'aser because Shemitah need not be eaten within the wall, and one suspected about Ma'aser is not suspected about Shemitah because Ma'aser can be redeemed, but Shemitah cannot. Why was this needed, if one suspected about a matter is not suspected about other matters, even if they are more lenient? Perhaps the Gemara could have said so, but since we find stringencies in each, it is better to answer this way. Alternatively, the Ramban said that one suspected of a matter is not suspected of something less severe only regarding different kinds of Isurim like what he discussed, i.e. cheese of Nochrim and oaths. He proved this from Orlah and Shechitah. However, Ma'aser, Shemitah and Taharos all involve food, therefore one suspected of a matter is suspected of something less severe. This answer is primary.

Rashba (Teshuvah 1:64): The questioner said that one suspected of a matter is suspected of something less severe. This is wrong. A Mumar to Orlos is not a Mumar to Shechitah. A Mumar to transgress one matter is a Mumar to the entire Torah only if he is a Mumar to idolatry or to be Mechalel Shabbos in public. However, for what he is a Mumar, he is a total Mumar, and he is not believed even with an oath.

Poskim

Shulchan Aruch (YD 119:5): One who is suspected of a severe Isur is suspected of an Isur with a lighter punishment, unless the latter is more severe in the eyes of people.

Prishah (9): One who is suspected about a severe Isur is suspected about a lighter Isur when he benefits from (e.g. sells) it. If he does not benefit from it, surely a Mumar to transgress one matter is not a Mumar (all) other matters, except for idolatry and one who is Mechalel Shabbos in public, like the Tur said above.

Rebuttal (Shach 13): There is no necessary to say that he is suspected only in a case in which he benefits. His proof is invalid. Indeed, only a Mumar to idolatry or to be Mechalel Shabbos in public is a Mumar to the entire Torah, but a Mumar to other Aveiros is suspected about matters more lenient than that Aveirah.

Taz (8): The Beis Yosef says that he is suspected about a lighter Isur only when both of them are the same kind of Isur, i.e. regarding food, but not regarding two kinds of Isurim.

Shach (12): The Tur says that one is suspected about a lighter Isur even if the punishment is not lighter, just it is lighter in people's eyes. It seems that he holds that if the punishment is lighter, he is suspected about even if it is more severe in people's eyes. If one does not fear Shamayim, all the more so he does not fear people (i.e. what people consider to be more severe). Rather, even if the punishment is not lighter, if it is lighter in people's eyes, since he is brazen to transgress in front of what people consider more severe, all the more so he will transgress what they consider light, even if it has a severe punishment. However, it seems that the Shulchan Aruch understood that that if the punishment is lighter, he is suspected about it even if it is equal in people's eyes. Further, even if the punishments are the same, if it is lighter in people's eyes, he is suspected about it. However, if he is suspected about a severe Isur, he is not suspected about a lighter Isur if it is more severe in people's eyes. People he fears people more than Shamayim. We can say that the Shulchan Aruch holds that one who is suspected about a matter is not suspected about something lighter in people's eyes if it has a harsher punishment, for perhaps he honors Shamayim more (than people). In order not to remove his Chezkas Kashrus, we judge him both ways leniently. Perhaps the Mechaber holds that it depends on what people are careful about. However, when matters are equal in people's eyes, then it depends on which has a greater punishment. This requires investigation.

Shach (8): The Ramban, Rashba and Beis Yosef hold that one is suspected about a lighter Isur only when both of them are the same kind of Isur. Also the Shulchan Aruch (Sa'if 2) says that a Mumar to Orlah is not a Mumar to Shechitah, based on Chulin 4b-5a. Even though the punishment for not circumcising is harsher than for (eating without) Shechitah, and it is more severe to people, since these are different Isurim, we do not say that one who is suspected about something stringent is suspected about what is more lenient. Perhaps this is why the Shulchan Aruch changed from the text of the Tur. The Tur said one who is suspected about a stringent matter..., and the Shulchan Aruch wrote 'one who is suspected about a stringent Isur', to teach that he is suspected only about a similar Isur that is more lenient.