A fraction field is not finitely generated over its subdomain

The fraction field $K$ has the structure of an $R$-module, and for any maximal ideal $\mathfrak{m}$ of $R$, it also naturally has the structure of an $R_\mathfrak{m}$-module. If $K$ is finitely generated as an $R$-module, then $K$ must also be finitely generated as an $R_\mathfrak{m}$-module because $R_\mathfrak{m}\supseteq R$. Thus we are reduced to the case when $R$ is local.

Let $R$ be a local domain with maximal ideal $\mathfrak{m} \neq 0$. Clearly, $\mathfrak{m}K=K$. But the Jacobson radical of $R$ is just $\mathfrak{m}$, so if $K$ is finitely generated as an $R$-module, then Nakayama's lemma implies that $K=0$, which is a contradiction.

If $K$ is finitely generated module over $R$, it is integral over $R$.
But given domains $R\subset K$ with $K$ integral over $R$, $K$ is a field if and only if $R$ is a field. (Atiyah-Macdonald, Prop 5.7)
So if $R$ is not a field but $K$ is, then $K$ is not integral over $R$ and thus not a finitely generated module over $R$.
Edit
It is perfectly possible for the fraction field $K$ of a domain $R$ that is not a field to be finitely generated as an algebra over $R$.
The simplest class of examples is that of discrete valuation rings.
Indeed if $R$ is such a ring and if $\pi$ is a generator of its unique maximal ideal , then we have $$K=R[\frac {1}{\pi}]$$.

Zev's proof uses the existence of maximal ideals, and is therefore not constructive. But it is not hard to make the proof constructive, and actually more generally:

Let $R$ be a ring and $R'$ a localization of $R$. If $R'$ is finitely generated and torsionfree over $R$, then $R=R'$.

Proof. Let $s \in R$ such that $s^{-1} \in R'$. Then $R' = sR'$. Generalized Nakayama's Lemma implies that there is some $a \in R$ such that $(1-as)R'=0$. Since $R'$ is torsionfree, this means $1=as$. So $R'$ has no more units than $R$, i.e. $R=R'$.