This is just a partial answer, but maybe an important case.
If $A$ is hermitian, $F(A)$ is the interval $[\lambda_{\text{min}}(A), \lambda_{\text{max}}(A)]$, where $\lambda_{\text{min}}(A)$ and $\lambda_{\text{max}}(A)$ are the minimum and maximum eigenvalues of $A$. Thus if $A$ and $B$ are both hermitian, $F(A+B) = F(A) + F(B)$ means $\lambda_{\text{min}}(A+B) = \lambda_{\text{min}}(A)+\lambda_{\text{min}}(B)$ and similarly for $\lambda_{\text{max}}$. That will happen if and only if $A$ and $B$ share a common eigenvector for their minimum eigenvalues and share a common eigenvector for their maximum eigenvalues.