Gemara

(R. Aba): Since one can permit his vow, it is like a Davar she'Yesh Lo Matirim, which is not Batel in a majority of Heter.

Question: One can annul a declaration of Terumah, yet it can be Batel!

Answer: Vows are different, for it is a Mitzvah to permit them. R. Noson taught that one who vows is like one who builds a Bamah. One who fulfills the vow (does not permit it) is like one who offered on the Bamah.

There is no Mitzvah to annul a declaration of Terumah.

Nedarim 52a - Mishnah: If someone forbade a food and it became mixed with other food, if the forbidden food can be tasted, the mixture is forbidden.

Yevamos 82a - Question: Why does the Seifa teach about a Tahor piece of Chatas mixed with Chulin? It should distinguish within the case of Tamei pieces!

Answer #3 (Rav Ashi): The Seifa discusses Davar she'Yesh Lo Matirim (it is permitted in another situation). Such a thing is never Batel, even if it is mixed with 1000 pieces.

This is wrong! For a Kohen, the piece is always permitted. For a Yisrael, it is always forbidden!

Rishonim

The Rif brings our Gemara and the Mishnah (a food forbidden by a Neder forbids a mixture b'Nosen Ta'am).

Ramban and Rosh (6:3): This is b'Eino Mino. B'Mino any amount forbids, for Nedarim are Davar she'Yesh Lo Matirim (59a), which is not Batel. The Yerushalmi suggests that Nedarim are Ein Lo Matirim because a Chacham uproots a Neder retroactively, as if it never was a Neder. A Neder that truly took effect is not permitted! It rejects this, and says that in any case the primary Heter is from now and onwards, for until now he was forbidden.

Rambam (Hilchos Nedarim 5:12): If one forbade certain wine to himself and one drop became mixed in a barrel it is all forbidden. Because he should permit his vow, it is like Davar she'Yesh Lo Matirim, which is not Batel b'Mino.

Lechem Mishneh (16): If Reuven forbade his food to Shimon, regarding Shimon it is Ein Lo Matirim because Shimon cannot permit the vow.

Question: Perush ha'Rosh (59a DH b'Terumah) says that Tamei Terumah in the hands of a Kohen is not Davar she'Yesh Lo Matirim because it is not common for the Yisrael to ask a Chacham to undo his declaration of Terumah. Tosfos (Avodah Zarah 63b DH Tevel) says that if the Yisrael is in the city, all agree that Terumah is called Yesh Lo Matirim!

Answer (Chasam Sofer YD 223): The Lechem Mishneh discussed one who forbade food only to Shimon. We cannot learn from one who forbade to everyone. Then, even if one person can permit it, it is called Yesh Lo Matirim for everyone. If something forbidden to Shimon and permitted to Reuven, the Lechem Mishneh teaches that it is called Yesh Lo Matirim only if Shimon can permit it. Poskim argue about something forbidden to everyone that later will be permitted only to some.

Or Some'ach (5:11 DH uv'Divrei): Even though the Rambam forbids Reuven to give the food to Shimon, we hold that Me'ilah applies to Konamos, and this applies only to Shimon. It is Ein Lo Matirim regarding Shimon. When it is Batel and permitted to Shimon, Reuven may give it to him.

R. Akiva Eiger (Pesakim 65): If someone vowed not to eat a Neveilah, even if he permits his vow, he may eat it only through Bitul. This is like an egg born to a Tereifah hen on Yom Tov. The Tzemach Tzedek (69) forbade it on Yom Tov like Davar she'Yesh Lo Matirim. Since after Yom Tov the Isur Nolad will go away, there is no reason to eat it on Yom Tov. Similarly, the Rambam says that Chametz b'Pesach is not Batel because it is Davar she'Yesh Lo Matirim. We rely on Bitul to leave over (not burn) mixtures of Chametz, for regarding Bal Yera'eh and Tashbisu it is Ein Lo Matirim. Nevertheless, it is not Batel regarding eating, for one may eat it after Pesach.

Rambam (Hilchos Ma'achalos Asuros 15:12): It seems that if Davar she'Yesh Lo Matirim was mixed with b'Eino Mino and did not give taste it is permitted.

Question (Ra'avad): (Why does the Rambam say 'It seems'?) A Mishnah explicitly teaches about this!

Kesef Mishneh: Perhaps the Ra'avad refers to the Mishnah (52a) that says that a food forbidden by a Neder forbids b'Nosen Ta'am. The Yerushalmi (Nedarim 21b) considers this Yesh Lo Matirim, and establishes the Mishnah to discuss Min b'Eino Mino.

Ran (59a DH Ela): Since it is a Mitzvah to permit vows, it is as if they were already permitted.

Poskim

Rema (YD 102:4): Davar she'Yesh Lo Matirim is only if it becomes permitted to the one to whom it is forbidden. If it will always be forbidden to Ploni, even if it is permitted to others, e.g. if he cooked food on Shabbos, it is Ein Lo Matirim.

Gra (15): The Rif rules like Levi, that we are not concerned for smell. If fish was baked in an oven with meat, a Beraisa forbids eating the fish with milk. This is even like Levi. It is Davar she'Yesh Lo Matirim, therefore we are stringent for any amount. Avi ha'Ezri says that this is Ein Lo Matirim, for what is forbidden (to eat it with milk) will always be forbidden. Terumas ha'Deshen (170) says similarly. They are correct, for Yevamos 82a concludes like them.

Rebuttal (Magen Avraham OC 318:2): Terumah (or Chatas) is different! It will always be permitted to Kohanim and forbidden to Zarim. What Ploni cooked on Shabbos is Yesh Lo Matirim for others (it will be permitted on Motza'ei Shabbos), so it is Yesh Lo Matirim even for Reuven! The Rambam considers Bikurim to have Matirim even for Yisraelim, even though they are always forbidden to them, because they have Matirim for Kohanim R. Shimshon disagrees. Perhaps the Rema discusses when it became mixed after Shabbos. Then it is just like Terumah (it is always forbidden to Reuven, and always permitted to everyone else). One should be stringent.

Chasam Sofer (YD 223): Perush ha'Rosh (59a DH bi'Terumah) says that Terumah given to a Kohen is not Davar she'Yesh Lo Matirim because it is rare to permit it. This does not contradict the Lechem Mishneh, who says that if the one forbidden cannot permit the Isur, it is Ein Lo Matirim. If something is forbidden to everyone and only one person can permit it, it is Yesh Lo Matirim for everyone. Meforshim (the Rambam and R. Shimshon) argue about something forbidden to everyone that will later be permitted only to some.

Shulchan Aruch (216:10): If one forbade certain wine to himself and one drop became mixed in a barrel it is all forbidden. Because he should permit his vow, it is like Davar she'Yesh Lo Matirim which is not Batel b'Mino.

Ha'Emek Shealah (132, p.110): R. Yanai holds that we are not Pose'ach with regret. The Gemara said according to him that a Neder is Yesh Lo Matirim because it is a Mitzvah to permit it. Usually, one finds a Pesach only if it is a Mitzvah to permit the vow. We hold that we are Pose'ach with regret, therefore a Neder is Yesh Lo Matirim even when there is no Mitzvah to permit it. The Rambam holds that we are Pose'ach Shevu'os with regret, therefore it is Davar she'Yesh Lo Matirim. We are stringent about Safek Tzedakah because it is Davar she'Yesh Lo Matirim. Even though it is not a Mitzvah to permit Nidrei Tzedakah, they are permitted through regret.