Obviously the Torah cannot enter the public domain (or even move 4 ammot within it) on Shabbat.
We know (Shemot 19:12) that when the Torah was given (which was on Shabbat) Moshe made sure to erect a boundary around Har Sinai making it a private domain (Rambam Shabbat 14:1). We also know that the encampment of the Jews in the desert is a classical public domain (Rashi Shabbat 6b).
The questions becomes, was there a valid eruv when the Torah was given? Generally, one cannot make an Eruv in a real public domain without doors (ShA OC 364:2). They didn't install doors because that would imminently create an obligation in Mezuzah which they didn't have any of yet. However, in this case God held a roof over the encampment (Bavli Shabbat 88a) in order to remove its status as a biblical public domain (OC 345:14) and make the Eruv Kosher.
However, we also know (Shemot 19:12) that no one was allowed to enter the mountain, giving it the Halachic status of Karpef which would prohibit carrying. This is supported by the Midrash which states that plants grew on Har Sinai just prior to the giving of the Torah. After all, a classical Karpef is freshly planted ground (cf OC 358:9-10).
At first glance, the prohibition of Karpef is removed because the walls surrounded Moshe's and God's dwelling on the top of the mountain. However, since the two of them only got there after the walls were put up (Shemot 19:20) they do not count as mechitzot shehukfu ledirah (OC 358:2). God (Shemot 19:23-24) even instructed Moshe to ensure that no one opened and closed the Eruv after he got up top, which would have made it a mechitza shehufah ledira (OC 358:2).
Thus the Eruv could only work to permit the Karpef if the area of Har Sinai was less than 5000 ammah^2. This was clearly not the case as the mountain covered the entire encampment, as above, including over 600,000 people, each of whom takes up 1 ammah^2 of surface area (Bavli Sukkah 7b "גברא באמתא יתיב a man sits in a square-ammah").
Yet we know God did give the Torah and it thus must have entered the public domain. We must conclude that the people were standing and not sitting. This is the source for the ancient custom of standing during the Torah reading of the 10 commandments.