Rabbi Moshe Sternbuch, chief of the Edah HaChareidis rules that Christian denominations that believe Yeshu ימש״ו is the embodiment of G-d chas v’shalom, fall under the category of actual Avodah Zarah and are not considered Shituf (link - link) "partnership".
“Christianity is not merely Shituf, which according to some opinions gentiles aren’t warned against, in truth they corporealize our G-d blessed be He, through their idolatry of the title “son”, the gravest form of idolatry." (Teshuvos VeHanagos siman 317)
(Since there is a prohibition against learning the ways of Avodah Zarah (Rambam, Avodah Zarah 2:2, Shulchan Aruch, Yoreh Deah 139) don’t open the links below without consulting a Rov.)
There is a common misconception amongst Yidden regarding mainstream Christianity, i.e. trinitarian Christianity. Their deity is not a sort of tritheism with G-d at its head. That would actually constitute heresy in trinitarian Churches. The Trinity is a “homoousion”, the belief that G-d is three distinct hypostases or persons who are coeternal, coequal, and indivisibly united in one being, or essence. In fact, that is why (link - link) they falsely claim to be monotheistic. In order to reconcile their three G-ds with the one G-d of the Torah they just combined them into one and called it a paradox. Even according to the opinions based on Tosafos to Sanhedrin 63b that shituf(1) is permitted for Bnei Noach, this doctrine should be actual Avodah Zarah. They don’t believe in some sort of partnership of false deities with the Creator or as intermediaries between them and Him. They redefined Him as composed of three primordial equal parts chas v’shalom, one even having form, i.e. Yoshke ימש"ו.
(1)One clear example of the definition of Shituf on Tosfos 63b “sh’mishtatfin shem shamayim v’davar acher”. The Rama OC 156 quotes Tosafos, the Olas Tomid on this Rama says:
“klomar sh’omrim d’ze l’mala mize (they recognize Hashem is above they’re AZ) veinam pogmim b’yichud Hashem (and they don't compromise Hashem's unity), omnam im pogmim b’yichud v’omrim sh’shnayim hem, ze bichlal AZ”.
Meaning if they believe it’s literally a partnership then it would be AZ (how much more so if they believe they’re one essence).
Others limit shituf even further (from an article by Rabbi Yaakov Goldstein): The form of idolatry prohibited for Nohadites is the belief that Hashem has completely left the earth, and plays no role in it. That it has been left completely to the authority of a deity to whom they pray and worship. In other words, they believe in G-d’s existence, but simply call him “Eloka Dielokaya/The G-d of G-d’s/.” This is prohibited for even a gentile to believe, and he must believe that G-d has direct influence on the world. It goes without saying that denial of G-d’s existence at all, and believe in a foreign deity, is defined as idolatry. However, Shituf, which is permitted for a gentile according to this opinion, believes Hashem interacts with the world, although has given authority to other deities or powers to also have some control in the world, and He is thus not the sole ruler. While Jews are prohibited from believing this due to idolatry, gentiles are not. (See Sefer Hamamarim Melukat 1:323, Mamar Mayim Rabim 5717 Melukat 3 ibid footnote 20)
In other words, they would be better off believing in a sort of tritheism than redefining G-d in this preposterous form (although as quoted above from Rav Sternbuch, by claiming Yoshke is the embodiment of G-d and not merely another deity, they are anyway corporealizing Him). Shituf is permitted according to some opinions, but only as long as they also believe in Hashem according to our definition. How is this doctrine any different from that of the Greek g-ds with Zeus as the chief deity of the pantheon with Yoshke instead? In conclusion, the Trinity is the Christian version of Zeus without any other of the sub-g-ds and is pure Avodah Zarah.
It should be noted though, the Lubavitcher Rebbe in a handwritten editing remarks to a letter
“Christianity is Avoda Zara, is in contrast to the seven Nohadite laws, as opposed to Islam. However, the Christians of today are simply “Maaseh Avoseihem Beyadeihem”.
Accordingly, the average layman may not be considered an idolator.
(Nontrinitarianism rejects the doctrine of the Trinity. The din of such churches depends on their respective beliefs)
(audio shiur on shituf based on Tosafos to Sanhedrin 63b)
האמונות והדעות לרבנו רב סעדיה גאון
תשובות והנהגות להרב משה שטרנבוך
האם הנוצרים בימינו עובדי עבודה זרה הם?