The short answer is no, people in Greenland do not live in igloos. Though sometimes you can see igloos in Greenland. Some weeks ago, there was an igloo building event in Nuuk, so at the moment you can experience igloos in town.
However it all makes sense, as the word igloo originates from the Greenlandic word for house "illu", which is pronounced more or less the same way. The igloo built with snow was used by the inuit people in Northern Greenland in the old days... but now they are mainly built for fun.
Do people even live in Greenland?
This summer when I was traveling from Iceland to the Faroe Islands I was chatting to the man sitting next to me on the plane. Of course we were talking about where we were going and where we came from. I told him that I am living in Greenland. He was very surprised - and told me, that he had never thought about, that people actually live in Greenland and just imagined it as a deserted place of eternal winter.
Greenland is huge compared to how many people live here and the distances between places are just as huge, there are however towns and villages along the coast, mainly along the west coast. As I already revealed we don't live in igloos, but especially the capital Nuuk, has quite a lot of different buildings. The typical picture of houses in Greenland is a lot of small colourful houses, but Nuuk is growing very fast and new buildings are popping up all the time. Apart from the colourful houses you will therefore also find many tall, modern apartment buildings. Many of the buildings have balconies and here you often see the mix of modern and traditional lifestyle, as it is quite normal to see dried fish, meat, reindeer antlers, and other things.
*I did unfortunately not participate in building the igloos in Nuuk, although I have participated in building and slept in an igloo in Northern Norway and I will present the proof beneath :)