What is the famous argument between the Vilna Gaon and the Ba'al HaTanya in understanding the Sod HaTzimtzum? And what are some ma'areh mekomos in their works where this Machlokes is seen?
There is a letter by the Lubavitcher Rebbe zt"l (original Hebrew text available online at chabadlibrary.org; an English translation is at chabad.org) in which he discusses this. (He also provides a list of places in Chabad Chassidic writings that talk about tzimtzum.) To summarize:
The two key variables here are:
(a) whether tzimtzum means "contraction" (i.e., something was there before and is not there now), or "concealment" (i.e., it's still there now as it was before).
(b) Whether tzimtzum affects G-d's own Self, or only the Divine energy ("light") that emanates from Him.
These two variables, then, yield four basic possible ways of understanding tzimtzum: as (1) the absence of G-d Himself from a certain space, or (2) of His light, or (3) the concealment of G-d's presence in a certain space, or (4) of His light.
The Vilna Gaon's view is that the approach #1 is the correct one: there are places in our physical world in which G-d is simply not present. The Baal Hatanya's view is the diametric opposite, #4: there is no place void of G-d's presence or of His light, but there are indeed places where it's not evident.
The Rebbe also notes that Nefesh HaChayim (by R' Chaim of Volozhin, one of the Vilna Gaon's star disciples) takes a middle ground, position #3.
The argument went much earlier.
There were students of the Arizal who held that Tzimtzum is literal.
For example, Yosher Levav (the author of the Mishnas Chassidim) wrote that it is based on both logic (that it is disgraceful for Hashem to be found in a dirty place) and because it is also what the Arizal taught him.
The Alter Rebbe (Baal Hatanya) disproves in Tanya (In Shaar Hayichud Vehaemuna) and says "It is possible to understand the error of certain scholars in their own eyes (May G‑d forgive them!)" which shows that they made a grave mistake in understanding Hashem's unity and are in need of atonement.
The Lubavitcher Rebbe said a Sicha in Nasso 5743 where he said that there were two legitimate ways of understanding tzimtzum (and says that the simple way to learn tzimtzum is like the Mishnas Chassidim!). Yet, once the law was decided (by both the Alter Rebbe and R' Chaim of Volozhin) that tzimtzum in not literal, nobody has the authority to argue on that decision.
This Mahlokes ultimately seems to stem from an earlier Mahlokes between Rabeinu Yosef Gikatelia (Shaarei Orah) and the Rashash (the Kabbalist). The SO holds that Keser of Atzilus (Keser Elyon) is the Ein Sof Himself and the Rashash holds that Keser of Atzilus is the Ohr of the Ein Sof.
If the K.E. is the Ohr E.S. then He Maintains His presence, but is Mitzamtzeim His Ohr. Those who say it is the E.S. Himself hold that He does withdraw Himself.
If you have no clue what I am referring to then this answer and the question that was asked is not for you.
I hope I'm understanding and presenting this correctly; someone please correct me if I'm getting this wrong.
I heard a talk from Rabbi Yaacov Haber about this. If I understand correctly, he contrasted the Tanya with a similar work by Ramchal known as "Adir BaMarom" or "Adir BiMromim." It's well-known that the Vilna Gaon thought highly of Ramchal.
"Tzimtzum" deals with the philosophical quandary that God is infinite, but He interacts with the physical world, which is finite. One side of the coin is that God "makes room" for anything else to exist, or else it wouldn't. The other side of the coin is that He allows some manifestation of Himself in the world as we know it. The question becomes, how real is that manifestation? Tanya believes in "tzimtzum gamur", absolute manifestation. Ramchal and the Vilna Gaon reject that.
If I recall correctly, Ramchal would say something to the effect of "every soul is Godly", but Tanya says "every soul is a piece of God" -- the danger here is the next step becoming "but some are more God than others." In light of the mess of Shabtai Zvi's messianic claims that were still in recent memory, the Vilna Gaon found this too dangerous.