Considering Iceland was mostly settled by Norwegians and Norway was Iceland's number one trading partner it must have lasted for quite some time but I can't find any sources that tell me for how long.
The Norse ages are divded into elder norse age (700 to 1050) and younger norse age (1050 to 1350), also called classical norse age. It was in this classical norse age that old Icelandic and old Norwegian split apart.
The old Norwegian language history can be split into 3 parts. From around 800 to around 1200 there is at the beginning no differences, but later small changes. In the other period (ca. 1200 - ca. 1350) there are few differences, but after that large differences from the older language comes. For example: You get normal ll, nn and ss, instead of rl, rn and rs. In front of f, p m l, r. The consonant i often changes to y.
The sources to the old Norwegian language history comes from two sources: Runes and handwritings. The rune inscriptions are almost all current time or a bit older than the old Norwegian litterature, but that's not very important in the language connection. There was one important and thorough handwriting from Karlevi in Öland, Sweden (around year 1000), one from Frösön, Jämtland, Sweden (around 1050), one from Flatdal, Telemark, Norway (around 1150) and one from Årdal, Sogn, Norway (around 1200).
In Norway runes for writing dissappered at the end of the 1300's, but in Iceland only after the reformation. But the latin alphabet had been taken in use already ca. 1050 in both Norway and Iceland. The alphabet had to change to fit the needs of the norse language. þ was kept from futhark, y and later ð was borrowed from anglosaxon, latin æ and œ , and modification of the latin letters with "branches" ę and ǫ and accents. The ortography in handwritings vary a lot, because there was no commonly accepted way of writing norse correctly.
Sources :