EDITED (my arithmetic being too hasty):

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).

At first sight I don't see an immediate reason for your proposed inclusion to be correct, though such an inclusion would presumably be "natural" when it exists.
Since the irreducible summands are uniquely determined in each tensor product, an inclusion would be the same as having the first list of summands included in the second list (multiplicity counted).  

In any case it's helpful to clarify the role of dual representations here.  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 $(\mu + \nu)^* = \mu^* + \nu^*$.  Putting this together, your inclusion would be equivalent to a statement that $V_\lambda \otimes V_\mu$ is included in $V_{\lambda+\nu} \otimes V_{\mu + \nu^*}$ for any dominant weight $\nu$.