Some posts [1] [2] suggest that the axiom of foundation (AoF) is essentially pointless, except that it allows us to prove that every set has a rank. I mostly deal with math built atop set theory, and I'm not familiar with any results that actually require (or are more easily proven by) the fact that every set has a rank.
Are any "important" theorems lost (or made harder to prove) by removing AoF? By important, I mean a result that is used in practice in mathematical theories built atop ZFC. For example, Zorn's lemma is important because it helps us prove that every vector space has a basis and every field has an algebraic closure. I'd like to know about similar applications of AoF, if they exist.
Consider the axiom scheme of collection, which says roughly $\forall x \exists y P(x, y) \implies \exists c \forall x \in B \exists y \in c P(x, y)$ for any predicate $P$.
I don't know of any way to prove collection from replacement without using foundation. The proof with foundation is:
For each $x$, let $\alpha_x$ be the smallest ordinal such that there is some $y$ of rank $\alpha$ such that $P(x, y)$. Then define $g(x) = \{y$ of rank $\alpha_x$ | $P(x, y)\}$. Then $\{g(x) | x \in B\}$ forms a set. Take its union $c$. Then $\forall x \in B$, $g(x)$ is nonempty and hence there is some $y \in g(x) \subseteq c$, and this $y$ must satisfy $P(x, y)$.
As mentioned in the comments, another application of foundation is showing that proper classes can be quotiented by equivalence relations.
Edit: according to this answer, collection does not follow from ZFC without foundation. So it's actually a necessary component of the proof.