Getting Biblical in Daily Life
Christian Perspective
St. Gregory of Nyssa—a saint in Catholicism too, but especially important to the Eastern Orthodox Church—said that he thought Isaiah, even though he lived so long before Jesus, really managed to capture the spirit of the Gospel: Isaiah "knew more perfectly than all others the mystery of the religion of the Gospel." That encapsulates not just the Eastern Orthodox attitude towards Isaiah, but also the attitude of almost all Christians, everywhere.
Most Christians see Isaiah as predicting—through the various prophecies about the Suffering Servant, Immanuel, the shoot from the House of Jesse—the coming of Jesus Christ. For Christians, it's the Hebrew Bible's version of Revelation, the book containing the most central prophecies about the nature and destiny of humanity, the coming of the Messiah, and the end of the world.
But it's not just the revered intellectual figures who've made use of Isaiah in Eastern Orthodoxy. Something called the "Dance of Isaiah" is a common feature at many Orthodox weddings. A song referencing how "Immanuel" will be born to a virgin is sung out, as the bride and groom walk with the priest around a table three times. (Here's a video.)
(If you want to learn more about the Eastern Orthodox Church's take on Isaiah, you can check out this webpage from the Orthodox Church in Australia, which argues for the position that Christ is the Suffering Servant described by Isaiah.)
What Would Jeromey-Rome Think?
St. Jerome—the great Catholic saint who first translated the Bible into Latin—had this to say about Isaiah, "He was more of an Evangelist than a Prophet, because he described all of the Mysteries of the Church of Christ so vividly that you would assume he was not prophesying about the future, but rather was composing a history of past events." So, that's pretty high praise—similar to the nice things ol' Greg of Nyssa had to say (see the Eastern Orthodox section above.)
The online Catholic Encyclopedia continues this tradition of admiring Isaiah. It's webpage on Isaiah discusses the esteem with which the different Church Fathers accorded Isaiah: "he successively assumes a mother's pleading and irresistible tone, and the stern manner of an implacable judge, now making use of delicate irony to bring home to his hearers what he would have them understand, and then pitilessly shattering their fondest illusions or wielding threats which strike like mighty thunderbolts."
Additionally, The Jehovah's Witnesses' official magazine—The Watchtower—takes its name from Isaiah, where the watchtower's look-out keeps an eye open for news that Babylon will fall.
(For more sources on Nyssa and Jerome, check out this and this).
Jewish Perspective
Isaiah is really important for Jews. He's one of the Major Prophets in the Hebrew Bible, after all. He helps simplify God's commandments, leading people to a purer understanding of what God wants. In fact, according to The Jewish Encyclopedia (an online copy of an encyclopedia published in 1906), Isaiah is second only to Moses among the prophets. He even in some ways was able to improve on Moses (being able to reduce the whole of the ten commandments into just two, according to rabbinical tradition: love and charity).
But Jewish tradition also records that God rebuked Isaiah for saying that he came from "a people of unclean lips." Even though God himself continually attacks his people for their sins, the same sentiments apparently sounded a little traitorous coming directly from Isaiah. He was making his people look bad in front of God, basically. Supposedly, Isaiah was killed by King Manasseh, which—in God's eyes, if not in the king's—was punishment for the unclean lips jibe. (Source.)
Chabad—the Orthodox Hasidic organization—pays tribute to Isaiah on their website, describing him as a prophet who portrayed the future and destiny of his people in unforgettable terms: "Isaiah's mission was not only to admonish the people to keep them on the right path. He also instilled fervent faith in G-d in the hearts of his flock, and he brought them courage and fortitude at a time when they were suffering mortal fear from the threat of the new Assyrian Empire. Isaiah also described in glowing terms the future glory of Zion, which inspires our people to the present day." (Source.)
Islamic Perspective
Isaiah is usually considered a prophet in Islam—although he isn't mentioned in the Qur'an or in the Hadith (the sayings of the prophet). But since the Qur'an says that there are plenty of prophets not mentioned in the Quran, there's always room to add one more to the list of acknowledged prophets (Sura 40:78).
There's another good reason for Islam to adopt Isaiah as one of its prophets: the Qur'an is very similar to Isaiah (and to the other major and minor prophets). It's not a narrative, but a collection of prophecies from different situations and times, arranged in a non-chronological way. Technically, Isaiah probably forms more of a narrative—there are chapters that take us back into the realm of historical events, framing the prophecies that Isaiah offers up—whereas the Qur'an is (for believing Muslims) entirely comprised of the direct word of God, beyond all narrative (though containing some stories).
The same prophecies that Christians read as pertaining to Jesus can also be read as pertaining to Muhammad. Moreover, Islamic tradition narrates some of the same stories about Isaiah found in Jewish sources, like the gory tale about Isaiah's death (he gets sawed to death while caught inside a tree—see the "Trivia" section for more info). (Source.)
Baha'i Perspective
Baha'is consider Bahuallah, the most recent "Manifestation of God" on earth, to be the "branch" foretold in Isaiah 11:1, which will grow and flourish throughout the land (it's the same figure Christians typically interpret as being Jesus). Thus Isaiah is a minor prophet for the Baha'is, but he isn't one of the big, heavy-hitters, the "Manifestations of God" like Jesus, Buddha, Muhammad, and some others. Bahaullah himself spoke about Isaiah, though, saying "I am the One Whom the tongue of Isaiah hath extolled." (Source.)