If $J_i$ is the angular momentum of the initial state and $L$ is the angular momentum of the $L$-pole radiation, then the final state must have angular momentum $J_f$ that is in the coupling of $J_i\otimes L$. Thus, for dipole $L=1$ transitions, we have $J_f=J_i+1, J_i$ or $J_i-1$ provided $J_i-1\ge 0$. For quadrupole $L=2$ transitions we have $J_i+2\le L_f \le \vert J_i-2\vert$.
Using this with $J_i=0$, one simply notes that $0\otimes L=L$, and is thus $J_f=0$ is never allowed.
The less mathematical explanation is that radiation from and $L$-pole transition carries $L$ units of angular momentum, so that, if you start with $J_i=0$, returning to $J_f=0$ when the radiation carries some angular momentum would violate conservation of angular momentum.