Gemara

Beraisa: After Shimon ha'Tzadik died, Berachah ceased from the Omer, Shtei ha'Lechem and Lechem ha'Panim. Each Kohen would receive only the size of a bean. The Tzenu'im (modest Kohanim) would refrain from taking, The gluttonous Kohanim took and ate.

Once, a Kohen grabbed his portion and that of another Kohen. They called him Ben Chamtzan (the robber) until the day he died.

Chulin 133a - Question (Rav Yosef): Is it proper to grab Matanos (from children bringing them to Kohanim) to show appreciation of the Mitzvah, or is this a disgrace?

Answer (Rava): It is wrong. "V'Nosan" - the Kohen should not take by himself.

Abaye (who was a Kohen): I used to take by myself. After I heard the above teaching, I stopped taking, but I would ask for Matanos Kehunah.

Beraisa - R. Meir: "Va'Yitu Acharei ha'Botza" - Shmuel's sons would ask for Ma'aser (and the verse degrades this)!

Abaye: After I heard this, I stopped asking, but I would take if people gave me.

Beraisa: The Tzenu'im would refrain from taking.

Abaye: After I heard this, I stopped taking. I take only on Erev Yom Kipur, so people will know that I am a Kohen.

Question: Giving Birkas Kohanim would suffice to publicize that he is a Kohen!

Answer: Abaye was unable to give Birkas Kohanim.

Rishonim

The Rif and Rosh (Chulin 45b and 10:7) bring the Gemara verbatim. They say that Abaye could not give Birkas Kohanim due to digestive problems

Rambam (Hilchos Bikurim 9:22): A Kohen should not grab Matanos Kehunah or ask for them. If it is given to him honorably, he accepts. When many Kohanim are in Beis ha'Mitbechayim, the Tzenu'im refrain and the gluttonous Kohanim take and eat. A Tzenu'a Kohen may take if he needs to publicize that he is a Kohen.

Poskim

Shulchan Aruch (YD 61:11): A Kohen should not grab Matanos Kehunah or ask for them. If they are given to him honorably, he accepts. When many Kohanim are in the kitchen, the Tzenu'im refrain and the gluttonous Kohanim take. A Tzenu'a Kohen may take if he needs to publicize that he is a Kohen.

Question: Tosfos (Chulin 133a DH l'Var) says that Abaye took only on Erev Yom Kipur because there were many Matanos and many Kohanim gathered to take, and if he would not take it would seem that he is not a Kohen. Why doesn't the Shulchan Aruch stipulate that one should refrain only if there are few Matanos?

Answer (Bach DH v'Ein Lo): The Rambam and Tur (and Shulchan Aruch) explain that Abaye took on Erev Yom Kipur because there were few Kohanim. Most did not come to take because Matanos were sent to their houses. If he did not take then it would seem that he is not a Kohen. The rest of the year there is no concern, for many Kohanim gather and people know that Tzenu'im refrain, like Tzenu'im refrained from Lechem ha'Panim.

Question: The Gemara concluded that a Tzenu'a Kohen need not take to publicize his Kehunah unless he cannot Duchan!

Answer (Bach, ibid.): People in Abaye's region knew that he was a Kohen, therefore it sufficed for him to take on Erev Yom Kipur so people will not assume that he is not a Kohen because he does not Duchan. However, if someone is not known to be a Kohen it does not suffice to Duchan, for some say that Birkas Kohanim is not a proof of proper lineage.

Question: Why didn't the (Tur and) Rashba (in Toras ha'Bayis ha'Katzar) distinguish between whether or not a Kohen can Duchan? A known Kohen must take occasionally only if he cannot Duchan!

Answer (Bach, ibid.): The Rashba holds that nowadays there is no Chacham who is so occupied with his Talmidim that he is unable to Duchan.

Question: The Tur proves from Lechem ha'Panim that Tzenu'im should refrain when many seek to receive. But Tzenu'im refrained only when the Berachah ceased from Lechem ha'Panim!

Answer (Drishah 3): Abaye had personal reasons not to take, i.e. to allow needier Kohanim to take, or because of "V'Sonei Matanos Yichyeh", or to avoid benefiting from people. Originally he thought that it is a Mitzvah to take Matanos Kehunah. He learned from Tzenu'im that it is not a Mitzvah, rather, a right to be supported by Yisraelim if the Kohen needs or wants. The Tur learns a different law from Lechem ha'Panim, that Tzenu'im should refrain when many seek to receive.

Rema (OC 53:22): A person should not be Shali'ach Tzibur without the consent of the Tzibur.

Source (Darchei Moshe 11): We learn from Yoma 39a that one should not quarrel about any Mitzvah.

Question (Chasam Sofer): Tosfos Yeshanim (DH she'Magi'o) says that if the Tzenu'im would have received a k'Zayis each they would have taken, for then there is a Mitzvah to eat it. Rashi (DH v'Chol) seems to agree.

Mishnah Berurah (65): One should not quarrel about a Mitzvah if it will be done. But if the Mitzvah will not be done, e.g. hosting guests or supporting a Chacham, one is obligated to protest to strengthen the Mitzvah. Letter of the law, we force people to do such Mitzvos.

Kaf ha'Chayim (96): It is detrimental for people with a Yartzite to quarrel about being Shali'ach Tzibur or Aliyos to the Torah. The best honor to the deceased and the greatest reward is for silence.

Rema (ibid.): If someone makes himself Shali'ach Tzibur by force we do not answer to his Berachos.

Source (Gra DH v'Chol Bava Kama 94a): We learn from "U'Votze'a Berech Ni'etz Hash-m"