Chabad is controversial because it challenges the mind to intellectually grasp the concept that the Atzmus (Essence of) Ein Sof (the Infinite Light) is present although concealed in every aspect of this physical world.
This above mention is idea can be explained through the concept of Tzimtzum. Is G-d's light contracted ie: removed from the world, or is it merely concealed, vested in creation with a purpose for the Jew to reveal it? The Lubavitcher Rebbe breaks it down to four ways of understanding (Source: Marbitzei Torah U’Mussar, Vol. III, p. 66.)
a) the tzimtzum should be interpreted literally, and moreover, that it affected G-d’s essence. The proof offered in defense of this theory is that it is impossible for the King to be found in a place of filth, Heaven forbid;
b) the tzimtzum should be interpreted literally, but that it affected only His light;
c) the tzimtzum should not be interpreted literally, but it affected the Source of light as well; and
d) the tzimtzum should not be interpreted literally, and it affected only His light.
A is followed by the Vilna Gaon and is the main maklokes between Chabad and Vilna.
Chaim Volozin actually holds by C and the Lubavitch Rebbe (while admits he has no proof) heard that R' Chaim was familiar with Chabad text and tried to find a resolution between Chabad and the Vilna Gaon.
Chabad follows option D and this is where the controversy begins.
Tzimtzum only effecting the light ie: revelation of G-dliness means that G-d can vest himself in this world and reveal himself through Jews, Torah, and Mitzvos.
This is the foundation of Chabad Chassidus, which is to make a Dira B'Tachtonim (Dwelling place below) for Hashem. This is sole purpose of creation and the concept has been foreign since the Churban and limited to small circles of kabbalists. Chabad revealed it to the world and this is the root of any and all controversy.
Torah Or, in the maamar Pasach Eliyahu Likkutei Torah, in the additions to Sefer Vayikra, the maamar Lehavin Mashekasuv beSefer Otzeros Chayim; Shaar HaYichud; Sefer HaMitzvos, [the maamarim entitled] Mitzvas Haamanas Elokus, and Shoresh Mitzvas HaTefillah (beginning ch. 34); in the Siddur, [the maamar] on the verse Zecher Rav Tovecho. In the maamarim of my revered father-in-law, the Rebbe shlita, that have been printed, [the subject is discussed] in [the maamarim entitled:] Al Yipater Adam 89, Dirshu Havayah 691, Al Kein Yomru 691, [Sichos] Shavuos 693, p. 8, and Tov Li 697.
The above all sources from this above sourced letter.
“It is not possible to ask a question about a [Rebbe being a] go-between, since this is G-d Himself, as He has clothed Himself in a human body.” (Likutei Sichos II:p. 510-511).
This controversial enough? The Tzaddik is a Chariot for Hashem. Avraham, Yitzhak, and Yaakov are all called chariots. (See Ch. 46 of Tanya) What is a chariot? It is something you get inside and control it to aid you in your bidding. But more importantly, a chariot is an object with no will of its own. It does exactly what its driver wants it to.