The short answer is... no, not at all. Tyson Yunkaporta is taking something extremely out of context, and it mostly happened after Mussolini's death.

Matera is a city in Italy with a cave system that had been used as housing since (likely) the Palaeolithic period. In the early 20th century, the caves were more or less a slum within the city, housing the poorest residents in abysmal conditions. In the immediate post-war period, however, the residents were relocated to new urban housing projects in order to "clean up" the city's image. There is an interesting writeup from the Smithsonian here.

Though the Matera caves had been used as dwellings since the Palaeolithic period, this was not a "Palaeolithic culture": residents did live amongst what could be considered ancient archaeological sites (murals, artefacts, etc.) but these were simply poor Italians. They really weren't living the kind of "caveman" image Yunkaporta is evoking. I should also note that "Cavernicoli" just means "cavemen"; it's not the name of a culture, a people, or anything similar. It was used pejoratively against the residents of Matera into the post-war period (see pgs. 120-121 here). I think Yunkaporta is taking one specific quote completely out of context: from the previous document, the Minister of Health Mario Cottellessa said at the time that residents were living in "primitive conditions of stone age cavemen".

In terms of what Mussolini had to do with this, there were some efforts to connect the caves to urban infrastructure and then to eventually relocate residents, but as Mussolini was ousted and hanged, he didn't live to see this. It wasn't until the 1950s that the "renovation" and relocation of residents occurred (put forward by the Christian Democrats). The last document linked talks about this entire renovation in great detail, mostly from a cultural perspective (the first couple of chapters should paint a good picture of the situation).

I've linked a couple of sources already, the first from the Smithsonian and the second a very hefty thesis from University College London, but if you're looking for lighter reading I'll throw in this New Yorker article about Matera's caves as well. Patrick McGauley has also written an academic history of the caves beginning in the immediate post-war period.

TL;DR There were people living in Palaeolithic caves in Italy, but they certainly didn't maintain a "Palaeolithic culture". Mussolini died before they were relocated from the caves. "Cavernicoli" just means cavemen in Italian, it isn't the name of a culture, and Yunkaporta is completely rewriting an actual historical event here.

Edit: highlighting a quote from the second source.