Under what conditions is the convolution of an input signal with the system's impulse response periodic?
I am somehow rephrasinge what @Hilmar already answered. In the continuous-time setting, convolution cannot invent frequencies. Indeed, there is an intimate relationship between Fourier and convolution, that is not always taught properly. Recently, Michael Bronstein was Deriving convolution from first principles

Have you ever wondered what is so special about convolution? In this
post, I derive the convolution from first principles and show that it
naturally emerges from translational symmetry.

The basic assumptions that a system is linear, and invariant by shift or in time (LTI) imply the concept of convolution. And convolution implies that a sine undergoing an LTI system will still be a sine, albeit with a time-shift (phase change) and amplitude modification (it can even vanish). In other symbols, through an LTI:
$$ \sin(\omega t)\to A\sin(\omega t+\phi)$$
$A$ can be zero, so frequencies can disappear. Yet if  $\sin(\omega t)$ is not present in the original data (no $\omega$), it cannot reappear. Frequency creation from nothing is a feature of nonlinear or time-variant systems.