In The Martian Chronicles, what happens to the Martian way of life? Is it completely destroyed?

The answer depends on which of the stories within The Martian Chronicles you're reading. At the beginning of the story, the Martian ways of life (there seem to be several) are vibrant ("The Earth Men," "The Third Expedition"), but the Martians become increasingly reclusive as more humans arrive ("-And The Moon Be Still As Bright," "The Martian"), to the point of barely resembling their former selves ("The Off Season"). By the end of the collection, all of the Martians are dead and their civilization is reduced to ruins, but there is a twist to this in the final page of "The Million-Year Picnic"; the human family, refugees from the war on Earth, are the true "Martians" now.
The Martians change somewhat from story to story in the early parts of the collection, mostly because Bradbury was writing these as short stories without a direct narrative link to each other. There's also the fact that the Martians are telepathic, and in the interests of pursuing an overall consistency we might argue that the "true" nature of the Martian way of life is never certain; not only are their minds inscrutable, but to truly know them would rob them of their alienness. This is perfectly demonstrated in "The Earth Men."
Perhaps the best exposition on what actually happens to the Martians is found in "-And The Moon Be Still As Bright" and "The Off Season." In the first story, it is revealed that the great majority of the Martians were killed by chicken pox contracted from the earlier (failed) expeditions. This is reminiscent of the fate of the Martians from The War of the Worlds as well as the fate of the indigenous peoples of the Americas during the early colonizations, when 90% of Native Americas died of smallpox contracted from settlers.
Later, in "The Off Season," the surviving Martians are still shown to be powerful telepaths, but now wear robes and masks and speak in near-riddles, and their emissary mentions "The Disease" and how he was sick for a long time, and one of the few survivors. Sam Parkhill is also mentioned to be passing deserted Martian cities as he flees them.
After "The Off Season" the Martians are not seen again, and it can be presumed that all of them have died, and their way of life has died with them.