Gemara

(Rav Yosef): Chachamim forbid Demai (produce bought from an Am ha'Aretz; it is a Safek whether or not it was tithed) before Kri'as Shem (declaring which parts are Ma'aser). R. Eliezer permits Demai before Kri'as Shem.

Abaye: All forbid Demai before Kri'as Shem. R. Eliezer does not suspect that an ignoramus did not separate Ma'aser Oni;

One (who does not want to give it away) may declare all his property Hefker. He is then an Oni, and he may (separate and) take the Ma'aser Oni himself.

Chachamim suspect that an ignoramus did not separate it. One is afraid to make his property Hefker, lest someone else take it!

Makos 16b (Rav Yosef): R. Eliezer and Chachamim argue about whether or not Ma'aser Oni is Tovel (forbids eating before Kri'as Shem).

Rejection (Abaye): If so, they should argue about whether or not one may eat (definite) Tevel before separating Ma'aser Oni!

Abaye: Rather, all hold that Ma'aser Oni of definite Tevel is Tovel. They argue about whether or not we are concerned lest an ignoramus did not separate Ma'aser Oni:

R. Eliezer is not concerned. Since Ma'aser Oni is permitted everywhere and to everyone, an ignoramus separates it and eats it himself. (Even though this is stealing from the poor, he avoids the Isur Tevel);

Chachamim are concerned. Perhaps he (did not know that a verbal declaration suffices and) did not want to toil to separate it.

Yoma 9a: Yochanan Kohen Gadol sent messengers throughout Eretz Yisrael and found that many people separated only Terumah Gedolah (but not Ma'aseros. He decreed that one must tithe Demai.) Therefore, Demai is exempt from Terumah Gedolah;

It is exempt from Ma'aser Rishon and Ma'aser Oni, for ha'Motzi me'Chavero Alav ha'Re'ayah (one who wants to collect, e.g. a Levi or Oni, must prove that he is entitled. Perhaps the seller already tithed it!)

Demai 4:3 (Mishnah - R. Eliezer): When tithing Demai, one need not be Korei Shem to Ma'aser Oni;

Chachamim say, he must be Korei Shem, but he need not physically separate it.

Sotah 48a: Yochanan Kohen Gadol told people 'just like there is Misah (b'Yedei Shamayim) for eating Terumah Gedolah (and its Tevel, i.e. produce from which Terumah Gedolah was not separated), there is Misah for Terumas Ma'aser and its Tevel.

Chulin 134a - Mishnah: If we are unsure whether a convert's cow was slaughtered before or after he converted, he is exempt from Matanos, for ha'Motzi me'Chavero Alav ha'Re'ayah.

Contradiction (Reish Lakish - Mishnah): If grain was found in an ant-hole in a place that was harvested, what rests on top (is Safek Leket. It) must be left for the poor. The owner keeps what is inside the holes (presumably, ants brought it there before the harvest);

R. Meir says, even what is inside must be left for the poor, because Safek Leket is Leket.

Answer (R. Yochanan): That Mishnah is the opinion of an individual (R. Yehudah ben Agra). The proper version is like our Mishnah (that exempts Safek Matanos);

Beraisa - R. Yehudah ben Agra citing R. Meir: Safek Leket is Leket. The same applies to Shichchah and Pe'ah.

Question (Reish Lakish): No matter who taught it, he has a powerful support!

Question: How do we explain "Oni va'Rash Hatzdiku"? It cannot mean to vindicate the poor - "V'Dal Lo Sehedar b'Rivo"!

Answer (Reish Lakish): Rather, give to him what (in other monetary cases) is yours (Safek Mamon, in which you are Muchzak).

Answer (Rava): Chazakah says that Leket must be left from the grain, but (until a Nochri converts) the cow is exempt from Matanos.

Rishonim

Rambam (Hilchos Ma'aser 9:1): Beis Din ha'Gadol found that everyone separated Terumah Gedolah, but Amei ha'Aretz did not separate Ma'aser Rishon, Sheni or Oni...

Rambam (2): One need separate only Terumas Ma'aser and Ma'aser Sheni from Demai. Chachamim required separating Terumas Ma'aser, for there is Misah for eating Tevel to Terumas Ma'aser. They required separating Ma'aser Sheni, for this causes no loss to the owner (he eats it in Yerushalayim). He need not separate Ma'aser Rishon or Ma'aser Oni, for we are unsure whether or not they were already separated. Ha'Motzi me'Chavero Alav ha'Re'ayah, i.e. the Levi or Oni cannot force him to give without proof that these tithes were not separated.

Rambam (3): Even though he need not physically separate Ma'aser Oni, he must`be Korei Shem. I.e. he says 'a tenth of what is here is Ma'aser Oni'. This is to fix Ma'aser Sheni, for Ma'aser Oni is given in the third and sixth years of Shemitah, and Ma'aser Sheni in the other years.

Question (Ri Korkus): The Rambam obligates Kri'as Shem only to habituate oneself, so that he will separate Ma'aser Sheni in other years. The Gemara (Makos 17a) says that Ma'aser Oni is Tovel, so Kri'as Shem is intrinsically necessary!

Answer (Ri Korkus): The Rambam holds that since most Amei ha'Aretz take Ma'aseros, we are concerned only for something for which (if it was not taken) one is Chayav Misah. There is no Misah for Tevel to Ma'aser Oni. Chachamim enacted to separate Ma'aser Sheni because it causes no loss (one may eat the produce). The Misah for Tevel to Ma'aser Rishon is only due to Terumas Ma'aser, therefore we do the minimum (Kri'as Shem) for Ma'aser Rishon that enables to properly separate Terumas Ma'aser. Rashi, R. Shimshon and Tosfos hold that there is an intrinsic need for Kri'as Shem for all Ma'aseros so the produce will not be Tevel. Perhaps they hold that there is Misah even for Ma'aser Oni.

Bartenura (Demai 4:3): R. Eliezer does not suspect that an Am ha'Aretz did not separate Ma'aser Oni. Chachamim say, even so one should separate it, for he does not lose.

Tosfos Yom Tov (DH Ein): The Bartenura connotes that also Chachamim do not suspect Amei ha'Aretz regarding Ma'aser Oni. They are stringent only because there is no loss. Perhaps he explains like the Rambam, like the Yerushalmi, unlike the Bavli. However, it appears that there is no argument. In the Yerushalmi. Chachamim address R. Eliezer according to his reasoning. Even if you do not suspect an Am ha'Aretz, there is no reason not to separate. The Bartenura could agree with this, but not the Rambam. It is not clear why the Rambam rules like the Yerushalmi against the Bavli.

Rebuttal (Mar'eh ha'Panim Demai 18a DH mi'Keivan): The Rambam (Ma'aser 9:1) explicitly says that Amei ha'Aretz did not separate Ma'aser Oni! The Yerushalmi explains that R. Eliezer trusts Amei ha'Aretz about Ma'aser Oni because he holds that there is Misah for its Tevel. Chachamim hold that there is no Misah for it, therefore they suspect Amei ha'Aretz about it.

Chazon Ish (Demai 1:4): The Rambam holds that Yochanan Kohen Gadol told people that there is Misah for Terumas Ma'aser and Tevel (to Ma'aseros) to make then zealous to separate it. Really, there is no Misah for Tevel to Ma'aser Oni.

Or Some'ach (Ma'aseros 9:3): The Rambam understands that Ma'aser Oni is Tovel because of "Lo Suchal Le'echol bi'Sh'arecha...; V'Ochlu bi'Sh'arecha v'Sove'u". Since Safek Ma'aser Oni is not given to the poor, it is not Tovel. Rav Yosef (Nedarim 84a) learns Tovas Hana'ah from "Kodoshav Lo Yihyeh"; R. Eliezer holds that Ma'aser Oni is not Tovel for it is not Kodesh, it is a mere monetary obligation. Since Safek Ma'aser Oni need not be given, one need not separate it; especially if (mid'Oraisa) Safek mid'Oraisa l'Kula (the Rambam's opinion). However, since often Ma'aser was not separated, Chachamim enacted to be Korei Shem to ensure that Ma'aser Sheni will be given in other years.

Rosh (Nedarim 84b DH v'Chachamim): One must be Korei Shem and say that the Ma'aser Oni is in the north or south. This is like Hafrashah.

Poskim

Shach (YD 259:14): Safek 'Hekdesh' for Aniyim or the Beis ha'Keneses is not considered Safek Isur about which we must be stringent. Rather, it is a monetary Safek, so ha'Motzi me'Chavero Alav ha'Re'ayah. Even though we are stringent about Safek Leket, Shichchah and Pe'ah (Chulin 134a), this is because the Chazakah was that the Matanos must be left. We always follow the Chazakah.