The Torah tells us that prior to the deeds perpetrated by the people of the Dor Haflagah who tried to build a giant tower in order to fight against Hashem, all the people on earth spoke one language (בראשית י"א:א'). Rashi, in his commentary on this Posuk (שם בד"ה שפה), indicates that this one language was Lashon HaKodesh, which we call Hebrew. This opinion is found as well in the Yerushalmi in Megillah (פרק א' הלכה ט, דף י.) where this universally spoken language is also identified as Lashon HaKodesh and as the language spoken by Hashem Himself. This latter point is a reference to the fact that Hashem created the world by speaking in Hebrew, as noted by the Pnei Moshe (שם בד"ה ואחרינא) and mentioned as well by Rashi earlier in the Torah (בראשית ב:כ"ג, בד"ה לזאת), citing the Midrash in Bereishis Rabbah (פרשה י"ח סימן ו'). It also refers to the fact that Hashem spoke to Bnai Yisrael in Hebrew when giving them the Torah, as noted by the Korban HaEidah (שם בד"ה בלשון), and stated as well by the Midrash (שם) and by the Gemara in Berachos (דף י"ג.) and in Sanhedrin (דף כ"א:). The Gemara in Chagigah (דף ט"ז.) adds that Hebrew is the language spoken by the Malachei HaShareis in Heaven.
Given this unique significance and status of the Hebrew language, is there any Mitzvah to study and master or speak Hebrew? The Yerushalmi in Shabbos (פרק א' הלכה ג', דף ט.) lists among the attributes which describe one who is guaranteed to be worthy of Olam Habba the fact that he speaks in Lashon HaKodesh; the Korban HaEidah (שם בד"ה ומדבר) notes that speaking this language leads to spiritual purity. This does not, however, mean that there is a Mitzvah to speak Hebrew. It is also obvious that knowledge of Hebrew and its grammatical and linguistic rules is sometimes necessary for proper understanding of an expression in the Torah which has Halachic ramifications, as is made clear, for example, in the Gemara in Yevamos (דף י"ג: ועיין שם בתוד"ה כיון) as well as in the Gemara in Rosh HaShanah (דף ג.), and as is elaborated upon by Rashi (שם בד"ה כדריש לקיש). This too, however, does not necessarily mean that there is a specific Mitzvah to speak Hebrew or that the study of Hebrew is even considered to be a fulfillment of the Mitzvah of Talmud Torah.
It would appear, however, that according to at least some authorities, there is some kind of Mitzvah associated with speaking and learning Hebrew. The Sifrei in Parshas Eikev (פיסקא י', דברים פיסקא מ"ו) states that when a child first begins to talk, his father should speak to him in Hebrew and teach him Torah, implying that this will guarantee the child a long life, and that failure to do so will unfortunately assure the opposite. The same idea is found in the Tosefta in Chagigah (פרק א' הלכה ג'), though with a slight variation: this source states that when a child knows how to talk, his father should teach him Hebrew. It could be argued that according to the latter source, it is insufficient to simply speak to the child in Hebrew, thereby familiarizing him with the language in a general sense; it is rather necessary to teach the child Hebrew so that he becomes fluent in it. In either case, it is clear that Chazal considered it important for children to be exposed to Hebrew at some level starting at a very young age; apparently there is value in knowing the language and, presumably, in being able to use it as an adult.
The clearest formulation which identifies learning Hebrew as a Mitzvah is found in the Peirush HaMishnayos of the Rambam, commenting on the Mishnah in Pirkei Avos (פרק ב' משנה א') which says that one must be as scrupulous regarding a "Mitzvah Kallah" - a minor Mitzvah - as one is with a "Mitzvah Chamurah" - a major Mitzvah. As an example of a Mitzvah Kallah, the Rambam (שם) cites studying - or teaching - Hebrew, along with rejoicing on Yom Tov (which is clearly a Mitzvah from the Torah), adding on, as the Mishnah itself (שם) seems to suggest, that these "minor" Mitzvos are in fact more important than people tend to think. The Rambam here clearly considers studying Hebrew to be a Mitzvah, one which is perhaps more significant than one may think.
The difficulty is that although the Rambam's view is clear in the Peirush HaMishnayos (שם), he does not codify this Mitzvah to learn or teach Hebrew in his Mishneh Torah, nor does such a requirement appear in the Shulchan Aruch. The Torah Temimah in Parshas Eikev (דברים י"א:י"ט, אות נ"ב) refers to a separate essay which he wrote about the obligation and the importance of learning Hebrew, and questions why the Poskim omitted any reference to the requirement to learn Hebrew. Although he suggests a possible answer, he concludes that the difficulty remains. It is worth noting, however, that among others, the Chavos Yair (שו"ת חות יאיר סימן קכ"ד) writes that it is important and indeed necessary to study Hebrew grammar, and the Vilna Gaon as well spoke of the need to be thoroughly familiar with grammar, as reported by his sons in their introduction to his commentary on the Shulchan Aruch (הקדמת בני הגר"א לשו"ע אורח חיים). Moreover, Rav Moshe Feinstein (שו"ת אגרות משה אבן העזר חלק ג' סימן ל"ה) actually states clearly that there is a Mitzvah to speak in Hebrew, although he asserts that there is certainly and obviously no prohibition to speak in any other language.
The Pardes Yosef in Parshas Ki Sissa (שמות ל:י"ג) quotes an interesting suggestion as to the source of this Mitzvah to study and know Hebrew, linking it with the Mitzvah of "Hakhel," a Mitzvah which obligated every Jew to assemble in Yerushalayim once every seven years (on the Sukkos following the Shemittah year) to hear the king publicly read certain sections of the Torah (דברים ל"א:י-י"ג). The Mishnah in Sotah (דף ל"ב.) says clearly that these sections had to be read by the king in Hebrew, a ruling codified by the Rambam (פרק ג' מהל' חגיגה הלכה ה'). The Gemara in Chagigah (דף ג.) implies that it was necessary for the people to understand what the king was reading; there is therefore a Mitzvah to learn Hebrew in order to properly fulfill the Mitzvah of Hakhel. One could suggest by extension that since the Torah and most other major Jewish works are written in Hebrew, there may be a Mitzvah to learn Hebrew in order to more thoroughly master these works, especially in view of the fact that the Ramban in Parshas Ki Sissa (שם) writes that Hebrew is in fact called Lashon HaKodesh precisely because it is the language used in the Torah and other holy works.
It is interesting to note that in the Shulchan Aruch, the Ramo (או"ח סימן ש"ז סעיף ט"ז) rules that whereas it is inappropriate to read certain types of stories, books, and literature on Shabbos, if they are written in Hebrew, they may be read on Shabbos. The Magen Avraham (שם ס"ק כ"ד) explains that this is because the language itself has Kedushah and one can learn Divrei Torah simply by reading books and even letters written in Hebrew. The Taz (שם ס"ק י"ג) disagrees with this last point, citing the fact that the Shulchan Aruch indeed rules elsewhere (או"ח סימן פ"ה סעיף ב') that one may speak in Hebrew about ordinary topics even in a place like a bathroom where Torah learning would be forbidden, but it is noteworthy that the Magen Avraham (שם ס"ק ב') quotes from the Sefer Chassidim (סימן תתקצ"ד) that it is a sign of piety to avoid speaking Hebrew in such places. The Midrash in Vayikra Rabbah (פרשה ל"ב סימן ה'), among other places, states that one of the meritorious deeds done by our ancestors in Mitzrayim was that they maintained their own language - Hebrew. Although this may not mean that they spoke exclusively in Hebrew, it is clear that they considered it important to know Hebrew fluently, and this was one of the things which made them worthy of redemption.