Gemara

Rav Asi: If one is from a place where people do not speak clearly, he may not Duchan (give Birkas Kohanim).

Support (Beraisa): If one is from a place where people pronounce 'Aleph' like 'Ayin' or vice-versa, he may not be Over Lifnei ha'Teivah.

R. Chiya (to R. Shimon bar Rebbi): If you were a Levi, your deep voice would disqualify you.

Rebbi (to R. Shimon): Tell R. Chiya (who pronounces 'Ches' like 'Hei') that when he reads "V'Chikisi la'Shem" (it sounds like 'v'Hikisi', I hit), he blasphemes!

Bava Metzi'a 85b: Eliyahu told Rebbi that the Tefilah of R. Chiya and his sons could bring Moshi'ach. Rebbi appointed R. Chiya to be Shali'ach Tzibur. Eliyahu needed to disrupt the Tefilah to prevent it from bringing Moshi'ach prematurely.

Rishonim

The Rif and Rosh (3:16) bring Rav Asi and the Beraisa.

Rambam (Hilchos Tefilah 8:12): We do not appoint to be Shali'ach Tzibur one who pronounces 'Aleph' like 'Ayin', or cannot pronounce the letters properly. A Rebbi may appoint one of his Talmidim to be Shali'ach Tzibur.

Kesef Mishneh: Even though normally we appoint to be Shali'ach Tzibur only one of the great Chachamim and Tzadikim of the Tzibur (Halachah 11), a Rebbi may appoint one of his Talmidim to be Shali'ach Tzibur.

Eshel Avraham (OC 53:15): The Rambam teaches about a Rebbi appointing a Talmid right after disqualifying one who mispronounces letters to teach that a Rebbi may appoint such a Talmid. He learns from Rebbi, who appointed R. Chiya.

Rambam (Hilchos Tefilah 15:1): Improper pronunciation disqualifies a Kohen from Birkas Kohanim. For example, one who pronounces 'Aleph' like 'Ayin' or vice-versa, or says 'Siboles' instead of "Shiboles", and similar mistakes. Likewise, if one speaks unclearly and not everyone can discern the words, he may not give Birkas Kohanim.

Kesef Mishneh: The Beraisa mentioned only one who confuses 'Aleph' and 'Ayin'. The Rambam understands that the same applies to 'Siboles' and "Shiboles", and one who speaks unclearly.

Question: Rebbi appointed R. Chiya to be Shali'ach Tzibur even though he could not pronounce 'Ches'!

Answer #1 (Tosfos Bava Metzi'a 85b DH Achtinhu): We do not appoint a Shali'ach Tzibur who cannot pronounce the letters properly when there is someone else who can. Here, there was no one else (who could bring Moshi'ach).

Answer #2 (Tosfos Megilah 24b DH keshe'Atah): When R. Chiya exerted himself he was able to pronounce it.

Rebuttal (Pri Chodosh 53:12): If so, why did Rebbi say that R. Chiya blasphemes when he reads "V'Chikisi la'Shem"? Surely, he would exert himself to say it properly! Rather, we must answer like Tosfos in Bava Metzi'a.

Answer #3 (Eshel Avraham ibid.): A Rebbi may appoint such a Talmid.

Poskim

Shulchan Aruch (OC 53:12): We do not appoint to be Shali'ach Tzibur one who pronounces 'Aleph' like 'Ayin' or vice-versa.

Magen Avraham (15): Ayin is stronger and deeper than Aleph. Likewise, we do not appoint one who pronounces 'Ches' like 'Hei'. If everyone speaks this way, he may be Shali'ach Tzibur.

Question: If someone confuses Ches and Hei, Shin and Samech, Kuf and Tes, or Reish and Dalet` how can he pray or read in the Torah? When he says verses such as "L'Nafsheinu Chiksah", he blasphemes!

Answer (Sefer Chasidim 18): Our Creator knows our intent. He only desires that we be wholehearted with Him. If the person does not know how to speak, Hash-m considers it as if he was Mekaven well. Similarly, Hash-m takes delight in people who sing Pesukei d'Zimra in a nice voice, even if they err because they do not know the verses - "V'Diglu Alai Ahavah". A case occurred, a Chacham told a Kohen to descend from the Duchan because he could not say the letters properly. The Chacham was shown from Shamayim that he will be punished if he does not return him.

Kaf ha'Chayim (61): If no one else is as great a Chacham and Chasid, this is considered that there is no one else.

Bi'ur Halachah (DH Ein): He should not be appointed even on occasion.

Mishnah Berurah (37): Likewise, if one speaks unclearly he may not be Shali'ach Tzibur (just like he may not give Birkas Kohanim). The Yerushalmi Berachos Sof 16b) permits 'Leshono Aruch'. Pnei Moshe (DH Im) says that he stammers, nevertheless he is permitted because he does not change the letters. It is easier to understand according to Sefer Charedim's Perush there (DH Ach), that this refers to one who pronounces the letters properly even though others of his city cannot.

Mishnah Berurah (38): The Mar'eh ha'Panim (Yerushalmi ibid., DH Lo) requires that he can pronounce properly, through exertion if necessary. The Pri Chodosh (ibid.) allows one who mispronounces letters when there is no one else as qualified. He would not allow a permanent appointment, for perhaps later someone qualified who pronounces them properly will come.

Kaf ha'Chayim (60): Birkei Yosef disallows one who can pronounce properly only through exertion, lest he will forget to exert himself.

Bi'ur Halachah (DH Mi): The Taz (128:30) is lenient (regarding Birkas Kohanim, and the same applies to Tefilah) because nowadays most of us do not know how to distinguish these letters.

Shulchan Aruch (128:33): One who does not know how to pronounce the letters properly, e.g. he pronounces 'Aleph' like 'Ayin' or vice-versa or similar mistakes, may not Duchan.

Mishnah Berurah (120, citing Taz 30): Nowadays most of us do not know how to distinguish them, so one is not disqualified for this. In Russia, even though people know the difference between Shin and Sin, since many do not distinguish them one is not disqualified for this, just like certain Mumim do not disqualify if Dash b'Iro (people are used to them).

Rebuttal (Mishbetzos Zahav 30): Mispronunciation distorts the Berachah, and distracts listeners. Dash b'Iro helps for the second problem, but not for the first. However, when the entire region speaks that way we are not concerned.

Kaf ha'Chayim (192): One should not rely on Dash b'Iro unless there is no else to Duchan. If someone else can Duchan, Kohanim who cannot should leave the Beis ha'Keneses before Retzei.

Kaf ha'Chayim (193): Poskim argue about whether or not a Kohen who mispronounces should go down from the Duchan if he already ascended. It seems that if Dash b'Iro, or if he can pronounce properly through exertion, he should not go down.

Magen Avraham (47): There is no Ayin in Birkas Kohanim. The Shulchan Aruch refers to one who does not distinguish Aleph from Ayin.

Rebuttal (Be'er Heitev): This does not answer the question! It could have said 'Aleph like Ayin', without mentioning vice-versa!

Defense (Kaf ha'Chayim 189): The Shulchan Aruch adopted the text of the Beraisa, which discusses Tefilah.

Eshel Avraham (47): There is an Ayin in the Birkas ha'Mitzvah beforehand (...v'Tziivanu Levarech Es Amo Yisrael b'Ahavah). Rashi (Megilah 24b DH Mipnei) says that pronouncing Ayin like Aleph harms Tefilah. Perhaps he is not concerned for the Berachah before Birkas Kohanim because one Kohen can say it on behalf of all the Kohanim.