Counterexample for the solvability of $-\Delta u = f$ for $f\in C^{0}$

You can write the solution as the convolution of Green's function with $f$. It will have the correct Laplacian (in some sense), but it will not necessarily have all second-order partial derivatives as continuous functions. The underlying reason is that inverting the Laplacian and then taking two derivatives amounts to applying a couple of Riesz transforms. These are singular integral operators of Calderón-Zygmund type, so they map $L^p$ into itself for $1<p<\infty$, and also map $C^{k,\alpha}$ into itself as long as $\alpha$ is not an integer.

At the integer exponents there is trouble, which already manifests itself in Calculus 2: the formula $\int x^p\,dx = \frac{x^{p+1}}{p+1}$ fails when $p=-1$. The power $x^{-1}$ is unique in that its integral is spread out evenly over all scales. This causes trouble in integral estimates: despite having expected control on each individual scale, we end up with a divergent integral anyway.

Other instances of this: Hilbert transform, harmonic conjugates. E.g., the conjugate of a harmonic function that is continuous on closed disk is not necessarily continuous. But the Hölder (and Dini) continuity is preserved under conjugation.

The problem is not the domain, the problem is that you are asking only $f\in C^0(\overline{\Omega})$. Take a look in Problem 4.9 of Gilbard-Trudinger.

Update 1: I found this example in the book of Qung Han, Fang Hua Lin - Elliptic pArtial Differential Equations (page 65):

Let $R<1$ and $B_R(0)=B_R$ the ball in $\mathbb{R}^N$ with center in origin. Let $x=(x_1,...,x_N)$ and define $$f(x)=\frac{x_2^2-x_1^2}{2|x|^2}\Bigg[\frac{N+2}{(-\log{|x|}^{1/2})}+\frac{1}{2(-\log{|x|})^{3/2}}\Bigg]$$

$$u(x)=(x_1^2-x_2^2)(-\log{|x|})^{1/2}$$

$$\phi(x)=\sqrt{-\log{R}}(x_1^2-x_2^2)$$

You can verify that $f\in C(\overline{B}_R)$, $u\in C(\overline{B}_R)\cap C^\infty (\overline{B}_R\setminus\{0\})$. Also,

$$ \left\{ \begin{array}{rl} \Delta u=f &\mbox{ in $B_R$} \\ u=\phi &\mbox{ in $\partial B_R$ } \end{array} \right. $$

Nonetheless, $\lim_{|x|\to 0} D_{11}u(x)=0$, which implies that $u\notin C^2(B_r)$.