What is a word that means "periodically purged" or "cleansed by fire"?
Swailing is the term used by the forestry community to describe the process of using intentional controlled burns to reduce the hazards of forest fires and to encourage new tree growth.
Usage of swailing appears particular to wooded parts of the UK (except, naturally, Scotland, where it is called muirburn or moorburn). Controlled burn is more common in the US. Prescribed burn is also common in Australia.
An elegant pen could make fine use of swailing in a literary context, but it may send readers running for a dictionary -- and, unfortunately, the reputable entries for swailing seem to be few.
The word salamanderize has been used with that sort of meaning (regenerate, rejuvenate), but I think in more recent usage just means to burn up without regeneration.
There are several mystical terms that reflect the concept, named with catch-phrases like fiery rebirth, Phoenix rising and star fire cycle. Less-mystical phrases include fire cycle (meaning the actual cycle of fire and regrowth in forests), rising from the ashes, tried in the crucible, and purified by fire. The latter two phrases appear in or stem from Proverbs 27:21 and Zechariah 13:8-9 respectively.
Serotiny is a related but plant-specific term: "an ecological adaptation exhibited by some seed plants, in which seed release occurs in response to an environmental trigger, [...] The most common and best studied trigger is fire". Etymology from wordnik: "Latin sērōtinus, coming late, from sērō, at a late hour, from sērus, late."
The term pyriscence refers in the main to seed release triggered by fire. Also (mis)used in reference to release due to dry conditions, which more properly is xeriscence.
Pyriscence looks like a word that ought to mean what you want, except it doesn't.