Is an infinite field always isomorphic to a non-trivial fraction field?

A field $K$ is the fraction field of a proper subring iff $K$ is not algebraic over $\mathbb{F}_p$ for some $p$. First, if $K$ is algebraic over $\mathbb{F}_p$, then every subring of $K$ is a field (since the subring generated by any single element is finite and a finite domain is a field), so $K$ cannot be the fraction field of a proper subring.

Conversely, if $K$ is not algebraic over $\mathbb{F}_p$ for any $p$, let $B$ be a transcendence basis for $K$ over the prime field and let $R$ be the subring of $K$ generated by $B$. Note that $R$ is not a field: if $K$ has characteristic $0$, this is because $R$ is a polynomial ring over $\mathbb{Z}$, and if $K$ has characteristic $p$, this is because $R$ is a polynomial ring over $\mathbb{F}_p$ in at least one variable since $B$ is nonempty by hypothesis. Let $S$ be the integral closure of $R$ in $K$. Since $K$ is algebraic over $R$, $K$ is the field of fractions of $S$. Since $S$ is integral over $R$ and $R$ is not a field, $S$ is not a field either.