Let $k$ be a finite field, let $X=\mathbb{N}\cup\{\infty\}$, and let $R$ be the ring of continuous functions $X\to k$ where $k$ has the discrete topology. (Equivalently, $R$ is the ring of eventually constant sequences of elements of $k$.) Note that ideals in $R$ are in bijection with closed subsets of $X$, by sending a closed subset $A\subseteq X$ to the ideal $I(A)$ of functions vanishing on it. Moreover, $R/I(A)$ can be identified with the ring of continuous functions $A\to k$. In particular, $R/I(A)$ is infinite iff $I(A)$ is infinite.
So, your question for this $R$ is equivalent to asking whether there is a minimal infinite closed subset of $X$. The answer is no, since for any infinite closed subset $A\subseteq X$, you can remove any point of $A$ besides $\infty$ to get a smaller infinite closed subset.
(More generally, you can get similar example with $X$ replaced by any infinite Stone space.)