Can anyone show an example of integral domain that is not integrally closed and also has one of its localization with respect to a maximal ideal not integrally closed?
Let $k$ be a field, and consider the domain $A = k[x,y]/(x^2-y^3)\cong k[t^2,t^3]$ (the isomorphism is given by $x\mapsto t^3$ and $y\mapsto t^2$). Note that $t\notin A$. The field of fractions of $k[t^2,t^3]$ contains $t = t^3/t^2$, but $t$ is integral over $k[t^2,t^3]$, since it satisfies the monic polynomial $z^2 - t^2$. Hence $A$ is not integrally closed in its field of fractions.
For the localization part of your question, you can localize at the maximal ideal $(t^2,t^3)$ in $k[t^2,t^3]$ (corresponding to $(x,y)$ in $k[x,y]/(x^2-y^3)$). The same argument as above shows that the result is not integrally closed. Note that if you try to localize at any other maximal ideal, you'll end up inverting $t^2$ and getting $t$ in the localization, so the argument will break down.