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The New Testament was the first book printed in Icelandic. It was translated by Oddur Gottskálksson (whose father was Norwegian) and published in 1540 (text here). Forty-four years later the whole Bible was printed in Icelandic thanks to Guðbrandur Þorláksson, a Protestant bishop at Hólar.
Afterwards, a number of other translations followed, such the Waysenhússbiblía, a reworking of a 1644 Bible from Hólar and printed by G. F. Risel in Copenhagen. In 1813, in the same city, the British Bible Society published the Grútarbiblía (or Hendersonsbiblía), just two years before the founding of the Icelandic Bible Society (Hið íslenzka Biblíufélag) by the British Bible Society's Ebenezer Henderson.
There came out a new translation in 1841, which was revised in 1863 by Professor (afterwards Bishop) Pétur Pétursson and Sigurður Melsteð, who compared it with the Greek and Hebrew originals and with the Norwegian, Danish, English and French versions. This edition (just the New Testament and Psalms) was edited by Eiríkr Magnusson and was reprinted in 1866, now as one volume with just the New Testament and Psalms and also another with the entire Old and New Testaments. Both the 1863 and 1866 editions were printed by the "Brit. Bible Soc." at Oxford, and this seems to be the translation in the illustrated New Testament of the Scripture Gift Mission in 1903 (London & Akureyri). By at least 1906, the British Bible Society (now printing in Reykjavík) put out a new New Testament translation based on the original texts, by Haraldur Níelsson, and by 1908 released the entire Bible.
The current publisher of the Icelandic Bible is the Icelandic Bible Society. In 1859 they printed the so-called Reykjavíkurbiblía, essentially the Viðeyjarbiblía from 18 years earlier. By at least 1899 they were printing the Old Testament translations of Þórir Þórðarson, which they used beyond the start of the 21st century. Their latest full translation, a new one by Guðrún Guðlaugsdóttir, was published in 2007.