The Tur (Orach Chaim 292) quotes a Medrash, which says that the reason we say, ואני תפלתי – “I am tefillah”, by Shabbos Mincha, is because we are coming to say, that even though we have been eating and drinking, we still turn to the Ribbono Shel Olam to daven. By the non-Jews, the more they eat and drink the more they get carried away, however, by Jews, even if we eat and drink, we still come to daven. A Kohen who drinks wine is disqualified from carrying out the avodah [Temple service], even if he drinks just one revi’is. Consequently, there is no obligation of וקדשתו, you shall sanctify him (the obligation to give priority to a Kohen), therefore, by Shabbos Mincha it’s ok to call up a Yisroel for the first aliyah.
However, the above is a tremendous chiddush as the Mishnah Berurah (201:13) quotes the Magen Avraham who learns that the din of וקדשתו is a din de’O’raisa. If, so, just because a Kohen has drunken a bit of wine and can’t perform the avodah, who says there is no longer any mitzvah of וקדשתו. Moreover, we find that if a Kohen is a ba’al mum [has a blemish] which is something that disqualifies him from carrying out the avodah on a permanent basis, there is still an obligation of וקדשתו (see Sha’ar HaTziyon 282:15, quoting R’ Akiva Eiger), so surely a Kohen who has had a bit of wine and is only temporarily disqualified should still have the obligation of וקדשתו.
The Maharsham asks: During Shacharis it’s also not possible to carry out the avodah, as we don’t have bigdei kehunah [the priestly garments]? The Maharsham then answers: The halachah is, מקריבים אע ״פ שאין בית – “nowadays we can offer up korbonos even though we don’t have a Beis HaMikdosh”, therefore, its technically possible to carry out the avodah if one would wear the bigdei kehunah, and be in the makom haMikdosh. However, a Kohen who has had wine, whatever happens, at this moment in time he can’t perform the avodah.
The Maharsham concludes: I feel that the main reason there would be room to allow calling up a Yisroel for the first aliyah of Shabbos Mincha is help keep peace. A Kohen shouldn’t forgo his right of being called up first, however, if occasionally he forgoes his right for reasons of peace it’s ok.
The Maharsham himself in Da’as Torah (135:3) takes on that the obligation of וקדשתו is de’O’raisa. See also Sefer HaMitzvos of the Rambam, Aseh 32; Sefer HaChinuch, Mitzvah 269, and Rashi in Sanhedrin 51b, d.h. v’hoyu kodesh.
The Shu”t Kesav Sofer (Orach Chaim 36) also writes: The reason a Kohen shouldn’t be mochel [forgiving] on getting the first aliyah is to avoid fights breaking out. If, however, by not being mochel, there will anyway be a fight, then it’s worthwhile for the Kohanim to act like Aharon HaKohen – lovers of peace, and be mochel on their kavod. (See Igros Moshe, Orach Chaim 2:34, and 3:20)