If $U^*DU=D=V^*DV$ for diagonal $D$, is $U^*DV$ diagonal too? All the matrices mentioned are complex $n\times n$ matrices. Let $U, V$ be unitary matrices such that $U^*DU=V^*DV=D$ for a diagonal matrix $D$ with nonnegative diagonal entries. Does this imply that $U^*DV$ is also diagonal? All I understand is that $U^*V$ will commute with $D$.

No. Consider $D=I$. Any unitary $U$ and $V$ satisfy the condition and yet it is easy to see $U^*V$ is not diagonal in general.
If the diagonal $D$ has all distinct diagonal entries, then $U$ and $V$ are both diagonal. Any kind of multiplication involving any of the three matrices is diagonal.
