Gemara

17b (R. Chiya - Beraisa): One is lashed mid'Oraisa for leaving the Techum without an Eruv.

Question (R. Yonason): [The Torah does not say 'Lo Yetzei Ish mi'Mkomo ba'Yom ha'Shevi'i', rather, "Al Yetzei..." One is not lashed for a Lav of "Al" [because this Lav also forbids Hotza'ah, for which one is Chayav Misah]!

Answer (Rav Ashi): It does not say 'Al Yotzi', rather, "Al Yetzei." The Lav forbids only Techumim. ("Lo Sa'aseh Chol Melachah" forbids Hotza'ah.)

43a - Question (Rav Chananyah): Does the Isur of Techumim apply more than 10 Tefachim above the ground?

Answer (Beraisa): If one said 'I will be a Nazir on the day that Mashi'ach will come', he may drink wine on Shabbos and Yom Tov, but he is forbidden on weekdays.

This is because Eliyahu will come "Lifnei Bo Yom Hash-m" (the day before), and we know [from tradition] that Eliyahu will not come on Erev Shabbos or Yom Tov, lest this disturb preparations [for Shabbos or Yom Tov].

Inference: Techumim do not apply above 10! If they applied, Eliyahu could not come on Shabbos, so Sunday would be permitted!

Rejection: The Tana is unsure whether or not they apply, so he is stringent. (If it does not apply, Eliyahu could come on Shabbos, and Mashi'ach could come on Sunday.)

45b (Beraisa): If rain fell on Yom Tov, one may take it wherever he may go.

Question: Perhaps it acquired Shevisah in the ocean, i.e. the rain came from clouds that formed on Yom Tov!

Answer: This is a doubt about a mid'Rabanan law, so we are lenient.

59a (Beraisa): Chachamim enacted Techumim to be stringent, and not to be lenient.

(Ravina): This means that Chachamim did not enact Techumim to be more lenient than the Torah, rather, to be more stringent. It is only mid'Rabanan.

Beitzah 36b (Mishnah): Whatever one is liable for on Shabbos due to Shevus, one is liable for it on Yom Tov, e.g. riding on an animal.

This is a decree lest one leave the Techum.

Question: [Since we decree due to this,] this implies that Techumim are mid'Oraisa!

Retraction: Rather, it is a decree lest one break a branch [to hit the animal].

Yerushalmi (Eruvin 4:6): [If one placed an Eruv inside the Techum, and later found it outside, it is Kosher.] This shows that Techum Shabbos is not mid'Oraisa.

Rav Mana says, this refers to the Techum of 2000 Amos, but the Techum of 4000 Amos is mid'Oraisa.

R. Shimon bar Karsena says, this refers to the Techum of 4000 Amos, but the Techum of 12 Mil is mid'Oraisa, like Machaneh Yisrael.

Rishonim

Rif (5a) and Rosh (1:24): Why is one lashed mid'Oraisa for Techumim? We hold unlike R. Akiva, who says that it is mid'Oraisa! The Yerushalmi brings opinions about whether mid'Oraisa the Techum is 4000 Amos, or 12 Mil, like Machaneh Yisrael. This shows that from 2000 Amos until 12 Mil one is lashed mid'Rabanan, and R. Akiva holds that one is lashed mid'Oraisa. All agree that one is lashed past 12 Mil. We learn from "Al Yetzei Ish mi'Mkomo." One may not leave his place, i.e. Machaneh Yisrael, which was 12 Mil. All permit 2000 Amos and below. According to the Bavli, only R. Akiva forbids mid'Oraisa. Why does R. Chiya's Beraisa say that one is lashed mid'Oraisa for leaving the Techum, i.e. 2000 Amos?

Rosh (ibid.): We can say that R. Chiya's Beraisa teaches that one is liable for Techumim mid'Oraisa, i.e. past 12 Mil, even according to Chachamim. Since there is some Techum mid'Oraisa, one is lashed mid'Rabanan even for the Techum of 2000 Amos, which is only mid'Rabanan. However, the Bavli holds that no Tana holds that Chachamim hold that there is Techumim mid'Oraisa, even for 12 Mil, like it says in Shabbos (69a) and Eruvin 45b.

Rambam (Hilchos Shabbos 27:3): It is a Safek whether or not Techumim apply above 10.

Magid Mishneh: The Rambam said that it is a Safek, but he did not explain how we rule. In a Teshuvah, he explained that we are stringent only when it is a Safek Torah. All Techumim in seas, rivers and Karmelis are mid'Rabanan.

Rosh (4:3): The question of Techumim above 10 Tefachim was not settled. We are stringent.

Hagahos Ashri: Or Zaru'a is lenient because Techumim are mid'Rabanan.

Tosfos (Chagigah 17b DH di'Chsiv): Rashi says that "u'Fanisa va'Boker" discusses Chol ha'Mo'ed, for one may not return home on Yom Tov, due to the Isur of Techumim. This is wrong. Only R. Akiva forbids Techumim mid'Oraisa, but Chachamim do not! Some say that Rabanan agree that 12 Mil is mid'Oraisa. The Yerushalmi says so, but the Bavli argues. If one forgot all the Melachos, we can say that he remembered that the Torah forbids Techumim, according to R. Akiva (Shabbos 69a). It did not say so according to Chachamim! In Beitzah 36b, the Gemara asked incredulously that our Mishnah teaches that Techumim is mid'Oraisa, and immediately retracted. This shows that Chachamim do not agree at all.

Tosfos (Pesachim 5b DH Lechalek): The Riva says that according to the opinion that burning is only a Lav, it is permitted on Yom Tov, since it is not called Melachah.

Chasam Sofer (7:59): What does Riva say about Mechamer or Techumim? Perhaps he permits Mechamer on Yom Tov. See OC 246:3, and Magen Avraham 12. (The Rema permits; the Magen Avraham says to be stringent like the Rif, Rambam and Beis Yosef, who forbid.) However, we do not find anyone who permits Techumim on Yom Tov. Surely it is forbidden, just like on Shabbos. This is clear from Beitzah 36b.

Poskim

Shulchan Aruch (OC 404:1): It is a Safek whether or not Techumim apply above 10. We are lenient about a Safek mid'Rabanan.

Beis Yosef (DH Yesh): Why is the Rosh stringent? He holds that it is a Safek mid'Rabanan even outside 12 Mil!

Kaf ha'Chayim (1): The Beis Yosef (495) brings from the Kolbo that on Yom Tov, the Torah forbids only Avos Melachos, and not Techumim. However, the Rashbatz (2:213) says that it is mid'Oraisa, and Aruch ha'Shulchan proves so from Beitzah.

Note: I did not find this in Aruch ha'Shulchan, but the Chasam Sofer brought this proof.

Rema: However, if he went like this outside of 12 Mil, we are stringent according to the opinion that Techumim of 12 Mil is mid'Oraisa.

Magen Avraham (1): It is mid'Oraisa only for the person. For his Kelim and animal, it is mid'Rabanan (Ralbach). It seems that Maharam Alashkar (41) disagrees.

Gra (DH l'Ma'an): The Rif (Eruvin 5a) brings from the Yerushalmi [that the Techum of 12 Mil is mid'Oraisa]. Tosfos (Chagigah 17b) brings such an opinion, but says that the Bavli disagrees. Another proof is from Shabbos 153b. It says according to R. Akiva [that there is an Isur Shabbos for which there is no Kares or Skilah, i.e. Techumim]. It did not say according to Chachamim. The Rosh and Ritva asked this, and from many places where the Gemara says that Techumim and Eruvin are mid'Rabanan. This is not a proof. Surely, 4000 Amos (i.e. counting both directions, or what is permitted through an Eruv - PF) is mid'Rabanan! However, there is a great proof from 45b. (It calls Techumim a Safek mid'Rabanan. Perhaps the water came from the ocean many Mil away!)

Damesek Eliezer: The Gra holds like Maharam Alashkar, cited in the Magen Avraham, that if Techumim is mid'Oraisa, it is not only for people. However, this is only for Kelim or animals of Yisrael. It does not apply to water in the ocean, which is Hefker! All agree that the Isur of something [of a Nochri] brought from outside the Techum is only mid'Rabanan! Also, the Rambam and Shulchan Aruch say that in oceans and rivers, even 12 Mil is only mid'Rabanan!

Note: Perhaps the Gra's proof is that we permit even more than 12 Mil from the ocean. Perhaps the clouds passed over 12 Mil of land, and this is called a Safek mid'Rabanan!

Shulchan Aruch (ibid.): All agree that there is no Torah Isur of Techumim in seas and rivers, since they are unlike the encampment in the Midbar.

Kaf ha'Chayim (15): If a place is not a full Reshus ha'Rabim mid'Oraisa, even if it is more than 12 Mil away, since the Torah permits carrying in all of it, it permits walking in all of it.