Assuming $A$ itself is time-independent. If $[A, U_t]=0$ for all $t$, then it can be proven that $[A, H]=0$: We have
$H=i \partial_t U_t\cdot U_t^\dagger$
Because $[A, U_t]=0$, it follows that $[A, \partial_t U_t]=0$ since we assume $\partial_t A=0$.