With $f$ and $g$ proper convex lower semicontinuous functions, consider the primal and dual problems $$\mathcal{S} = \arg\min_x f(x) + g(x)$$ $$\mathcal{S}^* = \arg\min_y f^*(y) + g^*(-y)$$
Fenchel-Rockafellar duality states that under the qualification condition $0 \in \mathrm{int} \, \mathrm{dom} \, f - \mathrm{dom} \, g$, $\mathcal{S}^*$ is not empty (and both infs are equal).
I'm looking for a counter example: functions $f$ and $g$ such that the dual has no solution (hence they must vioalte the QC).
In $\mathbb{R}^2$, I thought of taking $f$ and $g$ as indicator functions of the closed disks of center $(-1, 0)$ and $(1, 0)$ repsectively, both of radius 1 (they don't satisfy the QC and are an example of strict inclusion in $\partial f + \partial g \subset \partial(f + g)$). But it seems that even with these functions, the dual has a solution.
Define
$$ f(x) = \begin{cases} x\ln(x)-x, &\text{if $x>0$;}\\ 0, &\text{if $x=0$;}\\ +\infty, &\text{if $x<0$} \end{cases} $$ and set $g(x) := f(-x)$.
Because $f^*(y)=\exp(y) = g^*(-y)$, the dual is to minimize $2\exp$ which has no solution.
The primal has a unique solution: $\{0\}$.