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.

EDIT: It is **not** true that equality holds when $B = cA$, $c \in \mathbb C$.  For example, suppose $A = \pmatrix{0 & 0\cr 0 & 1\cr}$ and $B = i A$.  Then
$F(A) = [0,1]$, $F(B) = i[0,1]$, and $F(A)+F(B)$ is the convex hull of $0$, $1$, $i$ and $1+i$, but $F(A+B)$ is only the convex hull of $0$ and $1+i$.