Gemara

(Mishnah): Bi'ur (eradication) applies to Ma'aser and Bikurim, but not to Terumah.

Mishnah (Demai 1:2): Bi'ur does not apply to Demai (produce bought from an ignoramus. It is Safek Tevel; we separate Ma'aseros from it.)

Mishnah (Ma'aser Sheni 5:6): Bi'ur is on the first Erev Yom Tov of Pesach in the fourth and seventh years of Shemitah. One gives Terumah and Terumas Ma'aser to those who receive it (Kohanim)... In every place Ma'aser Sheni requires Bi'ur.

Beis Shamai say, a Tavshil (cooked dish) requires Bi'ur;

Beis Hillel exempt.

(Mishnah 7 - Beis Shamai): Nowadays, if one has produce when Bi'ur comes, he redeems it onto coins;

Beis Hillel say, coins and produce of Ma'aser Sheni are the same.

(Mishnah 8 - R. Yehudah): At first they used to alert people to fix produce (give the tithes) before Bi'ur, until R. Akiva taught that produce that did not reach Onas ha'Ma'aseros (was not yet obligated to be tithed) is exempt from Bi'ur.

Mishnah (10): At Minchah time on the last Yom Tov was Viduy... One says 'I eradicated from the house...'

Rishonim

Rambam (Hilchos Ma'aser Sheni 11:4): The obligation of Bi'ur and Viduy is whether or not the Mikdash stands.

Rebuttal (Ra'avad): The Viduy is only "in front of Hash-m", i.e. in the Mikdash.

Defense (Ri Korkus): The Sifri expounds that Viduy Bikurim is only in the Mikdash because it says "in front of Mizbach Hash-m". This implies that "in front of Hash-m" does not exclude when there is no Mikdash. The Shechinah is everywhere! Viduy Ma'aser in the Mikdash is only l'Chatchilah, for it is the primary place of Shechinah.

Mishnah Rishonah (on Ma'aser Sheni 5:7 DH u'Beis): According to the Ra'avad, also Bi'ur does not apply nowadays. Bi'ur is only to enable Viduy. The Rambam does not list Bi'ur to be a Mitzvah by itself, only Viduy.

Rebuttal (Chazon Ish Demai 2:3 DH bi'Yrushalmi): The Yerushalmi asked (why the Mishnah says to do Bi'ur), for Yochanan Kohen Gadol abolished Viduy Ma'aser! It answers that in any case one must do Bi'ur.

Rambam (7): The last Erev Yom Tov of Pesach (in the year after Ma'aser Oni applied) is Bi'ur.

Kesef Mishneh: Our text of the Mishnah says 'the first Erev Yom Tov'. (It is also the text of R. Shimshon and Tosfos ha'Rosh on the Mishnah, Tosfos (Rosh Hashanah 4a DH u'Ma'aseros) and others.) The Rambam's text is correct. The Yerushalmi asks why the Viduy was not on the first Yom Tov, and answers 'in order that people will have food to eat during the festival.'

Rebuttal (Gra YD 331:221, Shenos Eliyahu (Mishnah 6), and Perush on Yerushalmi 5:5): His proof for the Rambam is startling! We do not leave more than three meals after Bi'ur, like we find regarding Shemitah. The Yerushalmi would not suggest doing Viduy a week before Bi'ur! The correct text of the Yerushalmi suggests doing Bi'ur on the last Erev Yom Tov, right after Bi'ur, instead of on the following day. It answers 'in order that people will have food to eat on (the last day of) Yom Tov.'

Note: Presumably, the Kesef Mishneh explained like the Pnei Moshe (Yerushalmi Reish Halachah 5) that the Yerushalmi suggested doing Viduy on the first Yom Tov, i.e. and Bi'ur the day before.

Rambam (8): If one has produce of Vadai Ma'aser Sheni or Neta Revai (fruits of trees in their fourth year) or money used to redeem them, he gets rid of them and casts them to the sea or burns them. He need not do Bi'ur on Ma'aser Sheni of Demai.

Kesef Mishneh: The Mishnah mentions only Ma'aser Sheni, but we learn Neta Revai from Ma'aser Sheni through a Gezeirah Shavah. Also, Mishnah (10) explicitly includes Neta Revai with Ma'aser Sheni.

Radvaz: There are two kinds of Bi'ur. For Terumah and other Ma'aseros, which are given to others; their Bi'ur is to give them. Bi'ur for Ma'aser Sheni and Revai is to burn them or cast them to the sea.

Rambam (9): This is if he cannot eat them before Yom Tov will begin. If he has a Tavshil or Ma'aser Sheni or Revai he need not get rid of it, for a Tavshil is as if it was eradicated. The same applies to wine and spices.

Ri Korkus: The Yerushalmi says that all require Bi'ur of bread and oil, and all exempt spices and wine.

Rambam (10): Produce that did not reach Onas ha'Ma'aseros is exempt from Bi'ur.

Rosh (on Mishnah 5:7): Nowadays it is Midas Chasidus to redeem for the proper value. Letter of the law, one may redeem for a Perutah after the Churban.

Yachel Yisrael (23 DH uvi'Teshuvas): There is no Berachah for preventing an Isur, e.g. Bi'ur Ma'aseros or not eating Ever Min ha'Chai.

Poskim

Shulchan Aruch (YD 331:142): Bi'ur and Viduy of Ma'aseros apply whether or not the Mikdash stands.

Shulchan Aruch (144): One does not say Viduy until none of the Matanos remain by him. Bi'ur is on the last Erev Yom Tov. The Viduy is the next day. Produce that did not reach Onas ha'Ma'aseros does not inhibit Viduy. It is exempt from Bi'ur.

Question (Chazon Ish Demai 2:7 DH Tanan): Rishonim explain that Beis Hillel and Beis Shamai both require Bi'ur of Ma'aser Sheni. All agree that nowadays that we cannot eat it, leave it to rot at other times. Why is Zman Bi'ur different? Why do they argue only about nowadays? Even when the Mikdash stands, Beis Shamai should require redeeming Ma'aser Sheni outside Yerushalayim!

Answer (Chazon Ish): Beis Shamai do not obligate redeeming it before Zman Bi'ur or outside Yerushalayim, for perhaps the Mikdash will be rebuilt and he will take it to Yerushalayim and eat it. After it is buried, it is as if Bi'ur was done. Bi'ur is not an intrinsic Mitzvah. It is merely so Matanos will not remain in his possession. However, this does not apply to Tamei Ma'aser Sheni. Perhaps Beis Shamai say the same about Tamei Ma'aser Sheni even when the Mikdash stands.

Chazon Ish (DH v'Hayah): It would seem that nowadays Beis Hillel would not obligate Bi'ur, since the Mitzvah is to do with the produce what is proper so that none will remain at the time of Bi'ur. If he fails to do so, he must get rid of it. Nowadays there is nothing to do with the produce, so there is no Bi'ur. Beis Shamai hold that redeeming it is the Mitzvah. However, the Rambam and Rosh require Bi'ur even nowadays.

Chazon Ish (DH Nir'eh): One need not seek to get produce to fulfill Bi'ur. Perhaps it is Midas Chasidus, like burning Chametz on Erev Pesach. If one did not do Bi'ur the produce is not forbidden. However, one transgresses every day (that he leaves them in the world).