There are very few turf houses remaning in Iceland, even though everyone lived in turf houses in the past. Why were all these houses torn down?
In contrast, a lot more historic houses were preserved in the Faroe Islands (though these are typically wooden houses with a turf roof).
Good question!
For most of Iceland's history, turf houses made up an overwhelming majority of buildings in Iceland. Although trees were plentiful in the early stages of Iceland's settlement, the native birch is too crooked to be useful for construction of buildings, driftwood was available but in relatively small quantities. Too small to make entire structures out of.
A more plentiful material for construction was turf and stone. In the North of Iceland turf was generally used exclusively and layed in "bricks" that formed a sort of herringbone pattern.
In the south of Iceland, turf houses typically used a combination of turf and stone. Likely due to the south of Iceland being less vegetated than the North.
Turf houses were reasonably warm. The thick turf provided good insulation and the thick stone walls provided good temperature moderation. That being said, the gradual deforestation of Iceland and the resulting lack of firewood did force farmers in the country to change the way they built their homes. The viking longhouse gave way to "tunnel farms" by the late middle ages as the large open space of the longhouse became prohibitively expensive in wood to heat up as opposed to building separate chambers connected with tunnels and heating those with body heat.
The tunnel farms became the most widespread form of housing in Iceland up until the turn of the 20th century, which is when wooden houses and later concrete houses started to replace them at a rapid rate. In 1910 about half the population of Iceland lived in turf houses. By 1960 this figure had dropped to less than 1%. The last inhabited turf house in Iceland was abandoned in 1989.
The reasons for this are several:
Turf houses do not endure the elements very well. They were built without mortar and so the walls would start to sag over time. Freeze-thaw cycles would make the stones in the walls shift over the years and compromise their structural integrity. So turf houses were very maintenance intensive compared to their wooden and concrete successors. Walls had to be re-built every 20-50 years or so. They were also very prone to earthquake damage, particularly in the south of the country.
Urbanization meant that a large number of Icelanders, particularly poorer farmers, abandoned their rented plots and turf huts for Reykjavik and other sprawling towns in the country. The turf houses they left behind quickly crumbled.
Iceland rapidly developed in the late 19th until the mid 20th century. In a single lifetime most of the population shifted from subsistence farming to living in a modern, urbanised soviety. Turf houses were seen as backwards and their former residents did not particularly miss them. Many turf houses were bulldozed to make way for concrete structures. It wasn't until they had become rare that sentiments started to change. The Árbær turf house, one of the few original turf farms in the South of Iceland and the only one in Reykjavik, was abandoned in 1948 and left to rot for 8 years but renovated and turned into a museum in 1957.
I don't have a single source at hand, I'm a guide at the Reykjavik Open Air Museum and former warden at the Icelandic National Museum's Keldur turf farm. Much of the info I've posted can be found in the Icelandic language book Íslenzkir Þjóðhættir by Jónas Jónasson.
TL;DR: They are very difficult to maintain and Icelanders in the 20th century weren't fond of them so they let almost all of them decay or deliberately bulldozed them.