Is it a Mitzva to speak Loshon Kodesh? (sources)
The Rambam in Perush Hamishnayos Avos Perek 2:1 says that a Mitzva Kala is learning Loshon Kodesh.
Harav Yitzchak Yosef in Yalkut Yosef Hilchos Talmud Torah Seif Koton 78 also says it is a Mitzva.
See Igros Moshe Even Haezer 3:35 where he says it is a mitzvah to speak lashon hakodesh based of Sifri (Devarim Piska 46) which is quoted by Rashi on the verse of l'daber bam (Devarim 11:19). (The tshuvah is focused on non Jewish names.)
The Fifth Lubavitcher Rebbe writes that one should not speak Lashon Hakodesh as a day-to-day language.
As Lashon Hakodesh is a holy language, one shouldn't use it for mundane speech. Rabbi Shimon Ben Yochai said that if he would have been by Mt. Sinai, he would have asked for Jews to have another mouth, one to talk about one's needs and one with which to learn Torah. If one shouldn't talk with the same mouth Dvarim Bteilim and Torah, all the more so one shouldn't talk Davrim Beteilim in a holy language.
All the more so if someone will be talking forbidden words (Lashon Hara, lies, etc.) in a holy tongue. Moreover, it says that one is forbidden to put Lashon Hakodesh words into a song (even though songs aren't inherently something forbidden).
He explains the Yerushalmi (which says "He who lives in Israel, speaks Lashon Hakodesh, eats its fruit in purity and says Shma by day and by night is guaranteed a portion in the world to come.") as referring to one which possesses holiness of the place ("Lives in Israel"), holiness of the body ("eating the fruits"), and holiness of the soul ("speaks Lashon Hakodesh and says Krias Shma").
Moreover, he says that the common man only spoke Lashon Hakodesh during the first Temple (as they were very holy people). However, by the second Beis Hamikdash all commoners (as in non-Tanayim and Amorayim) spoke Aramaic.
Moreover, after the time of the Gemara, even Torah Scholars stopped using Lashon Hakodesh, as the spiritual level of the generation decreased.
Therefore, he says that one shouldn't speak Lashon Hakodesh as a day to day language.
The last Lubavitcher Rebbe points out, though, that nowadays modern Hebrew is no longer an issue as it absorbed enough non-Lashon Hakodesh words that it became just another language.
R. Qafih z"l published an article about this in Sinai (70: p. 197) entitled Anyone Who Does Not Speak Hebrew Correctly Violates a Negative Commandment (trans. my own). To summarize the article:
The Sifrei (Deut. 11:19) derives from the words “you shall to teach them to your children, that they speak them” that when one’s child begins speaking he should teach him Hebrew and Torah.
This does not appear to be a mere asmakhta (rhetorical flourish), but rather a halakhicaly empowered derasha, as implied by Tosefta (Hagigah 1:1) that a minor who knows how to shake, shakes the lulav, one who knows how to wrap himself is obligated in tsitsit, one who knows how to speak—his father teaches him Shema, Torah, and Hebrew. The juxtaposition of speaking Hebrew together with these other Biblical commandments implies that it too is a full-fledged obligation.
A very similar statement is found in the Sifrei Zuta (Numbers 15:38). In a similar vein Rambam writes in his commentary to the statement in Avot (2:1) that one should be careful with a minor mitzvah just like a major mitzvah, that examples of the latter are circumcision, tsitsit, ritual slaughter, and Passover. Examples of the former include rejoicing on holidays, and learning Hebrew. One again, the juxtaposition strongly implies that this is a fully binding obligation.
R. Qafih understands that according to this passage the obligation is Biblical. He explains Rambam’s omission of this in SHM and in MT as reflecting the view that the biblical commandment to master/speak/teach Hebrew (the three seem to get conflated) is included in the broader commandment to learn Torah. He adduces support from this from the Palestinian Talmud (Succah 3:12) whose text states that when a child learns to talk, his father teaches him the language of the Torah. The emphasis on Hebrew being the language of the Torah, implies that this study is auxiliary to the study of Torah.
Therefore, R. Qafih states that according to the Sifrei, Tosefta, and Rambam in PHM, anyone who does not learn Hebrew, or who learns it but does not speak it, violates a positive commandment, wheras according to the Palestinian Talmud, and Rambam in MT, the obligation to speak Hebrew is included in the obligation to learn Torah.
Furthermore, he cites R. Saadya Gaon’s commentary to Leviticus (10:1) that interprets Nehemiah (13:24) as referring to speaking a corruption of Hebrew intermixed with foreign languages as being improper; indeed, sinful. The next verse speaks of these individuals being struck.
Thus, R. Qafih concludes that according to Rasag one who speaks a corruption of Hebrew receives prophetically mandated lashes.
His concluding ruling is that anyone who does not learn Hebrew or who knows it but does not speak it violates a positive commandment, while anyone who speaks a corruption of it, or speaks without care for grammar, receives 39 lashes.
It is a mitzva to teach Lashon Hakodesh to one's children according to Rabbi Aaron Felder זצ"ל in Sh'elas Aharon, vol. 2 #21. In essence, in order to properly recite k'ri'as sh'ma, and understand the words of the Torah and t'fila, one must be a speaker of the language in which they are written. It does seem inconclusive on the point of whether this obligation pertains only to those living in the land of Israel [when the majority of Jews are living there too].
He bases this on an interpretation of the words of Sifri (D'varim 11:19 (46)) mentioned elsewhere, "when a child begins to speak his father teaches him Torah and speaks with him in lashon hakodesh". R' David Pardo explains that this statement of the Sifri assumes from the context in the Torah that we are talking about parents and children who live in the land of Israel and are therefore likely to be learning lashon hakodesh as their native tongue. Thus the main inference from the Torah's words is that of the first half of the statement - i.e. teaching Torah - because that is the part the father has a choice about. The need to speak to the child in lashon hakodesh is secondarily clued in by the otherwise nonessential words "to speak with them" (that is, the words that appear in the Torah), and by the threat of expulsion from the land that appears in neighboring p'sukim and would necessitate more effort to transmit the spoken (now foreign) language.
He goes on to counter the argument that if there really were such a requirement then it should have been included in the baraisa (Suka 42a) that lists things a father should teach his son. Since the baraisa was composed by people, place and time where lashon hakodesh was already the language of discourse, speaking it at home didn't need a mention.
Rabbi Felder's argument is consistent with his support in this same t'shuva of the opinion of Ramba"n (on Sh'mos 30:13) that by definition lashon hakodesh is the language in which the books of Tana"ch were composed. Thus the interchangeability of "Torah's words" with the spoken language in question and the de facto learning of one while learning the other.
There is no indication that one should speak lashon hakodesh exclusively, and indeed there is some discussion of sources descriptively or explicitly promoting the use of other languages, including the Sefer Hachasidim's observation of the prevalence of Aramaic for much of our history.
No. Source: Sefer Hachinuch. Generally following the Rambam, he lists all 613 mitzvos, and this is not one of them.