Have there been any applications of perfectoid theory to Iwasawa theory? At a first glance, this seems like a natural choice. For instance, the field $\mathbb Q_p(\mu_p^{1/p^\infty})$ is studied in both theories (the class group in Iwasawa theory and as a natural example of a perfectoid field (possibly after completion)).
Can we use the tilting correspondence to get information about class groups as in Iwasawa theory?
A flippant response is that people had the idea of using perfectoid theory in Iwasawa theory long before perfectoid theory even existed. What I'm referring to here is the work of Fontaine--Wintenberger, who studied the "field of norms" of a tower of p-adic fields, or the "tilt" as youngsters like you would call it, way back in the 1970's. Scholze's tilting equivalence generalises this to a much wider class of rings, of course, but for fields it's all there in Fontaine--Wintenberger.
Fontaine--Wintenberger's construction of the field of norms is the starting point for the theory of $(\varphi, \Gamma)$-modules, and these are essential tools for handling Iwasawa theory for p-adic Galois representations; see e.g. Cherbonnier and Colmez's JAMS paper "Theorie d'Iwasawa des representations p-adiques d'un corps local".
