Some synagogues do not have birkas kohanim (duchaning) on Yom Tov that falls on Shabbos. How did this idea arise?
The Beit Yosef says that the reason he heard that ashkenazim do not duchan daily is that the kohanim have the custom of going to the mikvah before saying birkat kohanim and it is too hard for them to toivel daily. The magen avraham says based on that opinion that one of the reasons that some communities do not duchan when yom tov falls out on shabbat is because there is a mitzvah to have relations with one's wife on friday night and then one would be a baal keri for birkat kohanim the next morning which would cancel the tevilla before yom tov so that he could duchan after going to the mikvah.
According to the Aruch Hashulchan, סימן קכח - כל דיני נשיאת כפים there is no reason for this Minhag.
אמנם זה שהיה מנהג מקדם דיום טוב שחל בשבת לא היו נושאים כפיהם – כבר נתבטל בימינו, כי אין במנהג זה טעם וריח, כמו שהאריכו מפרשי השולחן ערוך בזה (ט''ז סעיף קטן ל''ח, ומגן אברהם סעיף קטן ע').
He references the ט''ז and מגן אברהם as sources.
The ט''ז says:
- Some communities have the custom not to Duchen on Shabbat - and I see no reason for this - ואיני יודע שום טעם וריח - why should the day be worse, because it has a double Kedusha - Shabbat & Yom Tov?
- And it's appropriate to abolish this - וראוי לבטל דבר זה - and I saw one Gadol that commanded [the Cohanim] to Duchen On Shabbat-Yom-Tov.
The מגן אברהם says:
Some communities have the custom not to Duchen on Shabbat - and I see no reason for this - ולא ידעתי טעם לדבר - and possibly it's because of the of רבש"ע וכו - the [supplication added by the congregation towards the end of each of the 3 sections of Duchening starting with the words] Ribono Shel Olam.
While on Yom Tov we are more lenient with supplications (as we see from Avinu Malkeinu), we are stringent on Shabbat. So it would be best to abolish saying the רבש"ע supplication on Shabbat (rather than skipping Duchening).
Afterwards I saw in the Sefer פתרון חלומות that he saw in a תשובת הגאונים that one does not say the רבש"ע on Shabbat unless one is fasting for a bad dream. (He then argues on the Gaonim; if one can fast surely one can say the רבש"ע.)
So we see from the מגן אברהם a possible reason for the custom of not Duchening on Shabbat-Yom-Tov - because of the problem of saying supplications on Shabbat.
(We also see that he thinks it's better to skip the prayer than the Duchening. We also see that the at the time of the Gaonim they did Duchen on Shabbat.)