Sacred Language

There are some religions (either fictitious or not) that use a liturgic language for their rites and ceremonies. That language must be learned by all of their adepts, and sometimes it is even forbidden to translate the holy books to another language, mainly because it is said to have been created by God (or the gods), or to preserve its sacred euphony (good-sounding-ness).

Truth in Television, as it can be noticed below. See also Language of Magic and Black Speech.

Literature

 * In Elantris, the Derethi religion has Fjordell, language of the Empire where the Derethi church is centered, as its official language. At one point, Derethi high priest and Anti-Villain Hrathen debates with himself whether or not it's right to preach the religion to new converts in their native tongue, since Jaddeth (the Derethi god) revealed himself in Fjordell. Being the Magnificent Bastard that he is, Hrathen comes up with an elegant solution: Preach to people in their native language, then teach them Fjordell once they're converted.
 * This trope is parodied in another of Brandon Sanderson's works, The Alloy of Law, with the "High Speech" which the readers will recognize as the rather ridiculous sounding thieves' cant from the Mistborn trilogy.
 * In Anne McCaffrey's Acorna series, one planet considers Acorna a god because she speaks the native tongue of her people. They had been visited many many years ago by, and their priests had miraculously picked up the language in a single visit.
 * German philosopher Oswald Spengler speculated in his non-fiction book The Decline of the West that in the Stone Age, language (as in, speaking) may have been restricted to the priests, just as was the case with writing in some cultures.
 * The Lord of the Rings: Quenya appears to have been this for the Númenoreans, and possibly for others as well.
 * In the Firebird Trilogy, the Sentinels use Ehretan, the language of the world they originally came from, for their religious ceremonies, and one of their holy books is forbidden from being translated out of Ehretan.

Tabletop Games

 * Warhammer 40000 has Binary for tech priests.
 * According to the Warhammer Fantasy Roleplaying game, the priests of Sigmar learn the Dwarven language.
 * Dungeons and Dragons has the language called Celestial for the native tongue of the heavenly planes. For evil cultists, there is Abyssal and Infernal, the languages of demons and devils, respectively.

Druidic is also and example: all druids know Druidic, and if a druid teaches Druidic to a non-druid, they lose all their druid powers.

Video Games

 * Blood has the cultists speaking in "Domus Durbendia", a fake language similar to Sanskrit/Latin.
 * In the Dept Heaven series, Asgard has a special classical language that's referred to as the Sacred Tongue. While all characters from Asgard conveniently speak Japanese(/English) for the sake of the players and other characters, someone's fluency in the Sacred Tongue (which is rendered in Greek letters) is incontrovertible evidence that they are from Asgard. This is an early plot point in Riviera: The Promised Land, where Ursula's knowledge of the Sacred Tongue doesn't quite match up with the image Hector wants the main characters to have of her.

Real Life

 * Hebrew for Jews -- though the language has been updated for common speech in Israel, and a number of important Jewish texts are written in contemporary languages.

The Jews are by far the most serious about maintaining the sanctity of their religious language. A small but significant minority of Ultra-Orthodox Jews--particularly Hasids--regard the use of Hebrew for day-to-day conversation as blasphemous; even the ones who live in Israel refuse to learn Modern Hebrew and continue to communicate exclusively in Yiddish.
 * Arabic for Muslims. The translation of the Koran into other languages is permitted in Islam, but these translations are considered paraphrases or study aids, and not spiritually valid. This is related to it's roots as an oral tradition and Arabic's nature as a phonetic language. In fact, the written form of Arabic was originally formalized in part due to the need to record the sacred verses of the Koran permanently and all together.
 * Various sects differ on wether required prayers can or should be spoken in local vernacular or only in Arabic.
 * For over a thousand years (~500-1517 AD) Latin was the Sacred Language for western Christians, and remained so for Catholics up until the 1960s, and it technically still is the Sacred Language, even if few people use it. Every time the Pope releases a new encyclical it's originally in Latin, which is also Vatican City's official language.
 * Greek for eastern Christians.
 * Old Church Slavonic for the Russian Orthodox Church.
 * For Protestant Christians, Greek and Hebrew. The Old Testament was originally written in Hebrew and the New Testament in Greek.
 * Hinduism: The old Hindu vedas (scriptures) are written in Sanskrit. Pāṇini wrote on Sanskrit morphology and syntax for the sake of preserving the language as it was intended by the gods. He lived during the 4th century BC and is considered to be the first linguist.
 * Modern Hindi is this to Indo-Caribbean people in Guyana and Trinidad and Tobago. To clarify, the Indo-Caribbean people are descendants of indentured servants brought over by the British from India to the Caribbean. Over the years, Hindi stopped being the primary language, and everyone now speaks English (albeit a Caribbean dialect of it) as the primary language. Hindi only survives in religious texts and among the priesthood and temple singing groups, and your average Hindu in the Caribbean thinks of Hindi as a liturgical language, not something that one would speak in everyday life. People in India, of course, still speak Hindi as their most common language, and the linguistic differences between the Indians and the English-speaking Indo-Caribbeans often results in massive Internet Backdraft today when Indians accuse Indo-Caribbeans of "acting white" by speaking English as their primary language. The Indo-Caribbean people, for their part, retort that even though their ancestors were Indian, they now consider themselves to be a different--albeit related--culture.
 * Original Buddhism uses Pali/Maghadi (common people's version of Sanskrit).
 * Sumerian for the Akkadians and Babylonians. Historians record that the Akkadians, always fond of oddball religious practices, had at least one ritual involving two people whispering the same prayer into a bull's ears--Sumerian in one ear, Akkadian in the other.
 * Etruscan survived for some time as a ritual language after the language had "died". There was even a (now lost) Etruscan-Latin dictionary.