In some sense it did - the interior of Iceland is one of the most ecologically damaged landscapes in the world. But Iceland is a much bigger island than Rapa Nui, which may have given the settlers time to move and adapt, preventing total social collapse. Even so, Iceland did have periods of serious economic and social unrest which were probably connected to if not solely caused by ecological decline. Here's an answer to a related question from an actual Icelandic forester, u/StefnirSveinsson

Deforestation was just one of the problems that quickly became evident during the landnám or land-taking. Icelandic volcanic soil is renewed at a much slower rate than that of the Isles or the Scandinavian peninsulas. Traditional nordic agriculture could not be transplanted directly, and adaptations had to be made. Pigs were the first to go - hugely destructive, their rooting around literally turned pastures to mud and ash. Cattle and barley was not economical, but essential to maintaining traditional norse social structure - the big men had to hold feasts and serve beef and beer to their debtfolk to demonstrate their status and wealth, and milk was involved in pretty much every food preparation process. Even the greenlanders kept cattle, though the marginal conditions meant they ended up breeding a kind of miniature cow. Imports paid for by exporting wool, and later dried cod covered the extra demand.

Sheep grazing was made sustainable by a complex system of grazing rights, which we see archaeological and historical signs of in the 1100's to 1200's, but also made sure the forests were not allowed to regenerate, halting but not reversing any of the damage.