I assume you are working over a field like $\mathbb{C}$ of characteristic 0, where it's equivalent to work with the Lie algebra (and in any case $G$ might as well be taken to be semisimple).
A somewhat random example for type $A_2$ seems to show how your proposed inclusion will typically fail. First, recall that $V_\mu^*$ is itself irreducible of highest weight $\mu^*:= -w_\circ \mu$ (where $w_\circ$ is the longest element of the Weyl group). Moreover, on the right side you have $(\mu + \nu)^* = \mu^* + \nu^*$. In type $A_2$, a dominant weight abbreviated by $(r,s)$ with $r,s \in \mathbb{Z}^+$ has "dual" weight $(s,r)$. Now take $\nu = (0,1)$ and consider the special case of your question: $$V_{(1,0)} \otimes V_{(2,0)} \hookrightarrow V_{(1,1)} \otimes V_{(1,2)} ?$$
Using Brauer's standard method for decomposing the tensor product on the right side, you don't get the irreducible $V_{(3,0)}$ as a summand, which obviously occurs as a summand on the left side.