It is not true that $\text{rank}_+^*(V) \leq \text{rank}_+(V) $. In fact, an equivalent definition of the non-negative rank of $V$ is the minimum number of non-negative vectors (not necessarily columns of $V$) such that every column of $V$ is a conic combination of these vectors. Therefore, the opposite inequality $\text{rank}_+^*(V) \geq \text{rank}_+(V) $ holds.
For an explicit example where $\text{rank}_+^*(V) > \text{rank}_+(V) $, consider the matrix
$$ V= \begin{bmatrix} 1 & 0 & 1 & 2 \\ 0 & 1 & 2 & 1 \\ 0 & 0 & 1 & 1 \end{bmatrix} $$
It is easy to see that $\text{rank}_+^*(V) =4 $ and $\text{rank}_+(V)=3$.