The שולחן ערוך או"ח קכח:לג, based on the :גמרא מגילה כ"ד states that someone who cannot properly pronounce the letters, such as one who mispronounces an א as an ע or an ע as an א, should not go up to perform ברכת כהנים. The difficulty with this is that the פסוקים do contain an א but do not contain an ע. Why then would someone who mispronounces an ayin as an aleph be forbidden from performing ברכת כהנים?
רש"י in the גמרא seems to be sensitive to this issue. He gives a specific example of a grievous mispronunciation that would result with the exchange of an ע for an א. However, when explaining the opposite substitution, he writes simply that as a result of this substitution he will disqualify his prayers. This statement of רש"י is quite vague and requires further interpretation but it shows, nevertheless, that רש"י addressed the lack of an ע in ברכת כהנים.
The issue is dealt with further in the commentaries on the שולחן ערוך. The question is raised in באר היטב but no answer is given. מחצית השקל seems to suggest that this is not an issue as the גמרא is simply referring to one who confuses the two letters. Thus, as long as one of the substitutions is significant, it is a sufficient problem.
I suggest a possible explanation for the גמרא which may be the meaning of רש"י as well. After the כהנים complete the main part of ברכת כהנים, they recite an additional prayer which begins, "רבונו של עולם, עשינו מה שגזרת עלינו..." concluding with the פסוק:
הַשְׁקִיפָה מִמְּעוֹן קָדְשְׁךָ מִן-הַשָּׁמַיִם, וּבָרֵךְ אֶת-עַמְּךָ אֶת-יִשְׂרָאֵל, וְאֵת הָאֲדָמָה, אֲשֶׁר נָתַתָּה לָנוּ--כַּאֲשֶׁר נִשְׁבַּעְתָּ לַאֲבֹתֵינוּ, אֶרֶץ זָבַת חָלָב וּדְבָשׁ
Pronouncing the ע of נשבעת as an א would drastically change the meaning from "you have sworn" to sounding something like נִשְׁבָּתָּ, "you have been taken captive." Although this is not part of the actual blessings of the כהנים, perhaps it is a serious enough mispronunciation to forbid a כהן from performing ברכת כהנים.
Another suggestion made by Snag in the comments is that the real שם המפורש used in the בית המקדש might contain an ע. But that doesn't fit with רש"י and also doesn't account for why we would still be particular about this today.
Another suggestion made by Snag in the comments is that the real שם המפורש used in the בית המקדש might contain an ע. But that doesn't fit with רש"י and also doesn't account for why we would still be particular about this today.
