Here is a proof that this is impossible for commutative rings larger than the continuum or finitely generated maximal ideals. Let $A$ be an uncountable commutative ring and let $I\subset A$ be a countable maximal ideal. Then $A$ acts on $I$ by multiplication, giving a homomorphim $\alpha:A\to End_A(I)$. If either $A$ is larger than the continuum or $I$ is finitely generated, $A$ will have larger cardinality than $End_A(I)$. In either case, we can conclude that the kernel of $\alpha$ is uncountable.
In particular, we can find some $k\in \ker(\alpha)\setminus I$. Now by maximality of $I$, there is some $a\in A$ and $i\in I$ such that $ak=1-i$. But then for any $j\in I$, $0=akj=j-ij$. This implies $i$ is an idempotent generator of $I$.
Unfortunately, I don't see any obvious way to get rid of the extra hypotheses in this argument. It is possible for an uncountable ring to act faithfully on a countable ideal; consider $A=\mathbb{Q}^\mathbb{N}$ and $I$ the ideal of sequences that are eventually $0$. Of course, in this case $I$ fails to be maximal.
EDIT: Here is an example, which can have any cardinality up to the continuum. Choose an uncountable algebraically independent set $S\subset \mathbb{R}$, and let $k=\mathbb{Q}(S)$. For each $x\in S$, choose a sequence $(q_n(x))$ of rationals converging to $x$. Given any $y\in k$, write $y=f(x_1,\dots,x_m)$ for some rational function $f$ and some $x_i\in S$ and define $q_n(y)=f(q_n(x_1),\dots,q_n(x_m))$. Note that $q_n(y)$ may not be defined since the denominator of $f$ could vanish, but for fixed $y$ it is defined for all sufficiently large $n$ since $q_n(x_i)$ must converge to $x_i$.
Now let $A$ be the set of all sequences of rationals $(a_n)$ such that for some $y\in k$, $a_n=q_n(y)$ for sufficiently large $n$. This set has the same cardinality as $S$, and forms a ring under pointwise addition and multiplication. There is an epimorphism $\varphi:A\to k$ that sends any sequence to its limit. The kernel of $\varphi$ is a countable maximal ideal of $A$. Explicitly, $\ker(\varphi)$ is the set of sequences of rationals that are eventually $0$.