Disjoint ROCs of input and system function
If the ROCs don't overlap, the corresponding convolution sum doesn't converge, hence the output is undefined. Note that in your question it is assumed that the $\mathcal{Z}$-transforms of the input as well as of the system's impulse response both exist.
You can construct examples for which neither the $\mathcal{Z}$-transform of the input nor of the output exist, yet the input and output signals are perfectly defined. E.g., the input sequence $x[n]=A\cos(\omega_0n)$ has no $\mathcal{Z}$-transform, neither has the corresponding output sequence $y[n]=B\cos(\omega_0n+\phi)$, where $B$ and $\phi$ depend on the system's frequency response. [In this example it is assumed that the ROC of the system's transfer function $H(z)$ includes the unit circle, i.e. the system is stable.]