The answer might depend on how one defines the monoidal structure on the realizations $|-|:s\mathcal{V}_i\to \mathcal{V}_i$. In the topological and the simplicial cases the strong monoidal structures are determined by the preservation of products. 

Observe that in both of these cases the canonical morphism $|X\times Y| \rightarrow |X|\times|Y|$ is induced through the diagonals $\Delta^n \rightarrow \Delta^n\times\Delta^n$ when one thinks of it as a map between the coends. One can generalize to the monoidal situation by requiring the cosimplicial object $\Delta \to \mathcal{V}$ to be a comonoid in $[\Delta, \mathcal{V}]$, so that we have $\delta_{\Delta^n} : \Delta^n \rightarrow \Delta^n\otimes\Delta^n$. This will define a colax monoidal structure on $|-|$

$$\int^n X_n\otimes Y_n \otimes \Delta^n \rightarrow \int^{n,m} X_n\otimes Y_m \otimes \Delta^n\otimes \Delta^m$$
$$\cong (\int^n X_n \otimes\Delta^n) \otimes (\int^m Y_m \otimes \Delta^m)$$

We also need $\otimes$ to commute with coends. 

If one is given a lax monoidal functor $F$, then the composites $F|-|$ and $|F-|$ are neither lax nor colax, since $|-|$ is colax. However, one can still form the diagrams for $\tau$ by inverting in them the monoidality stucture maps for $|-|$. If $\delta_{F\Delta^n}$ equals to $\delta_{\Delta^n}$ up to the isomorphism $F(\Delta^n) \cong \Delta^n$, then these diagram will commute. The first diagram will commutive since (writing $.$ for the tensor, and omitting an index $n$)

$$FX.FY.F\Delta \rightarrow{} FX.FY.F\Delta.F\Delta \rightarrow F(X.\Delta).F(Y.\Delta) \rightarrow F(X.\Delta.Y.\Delta)$$
$$=$$
$$FX.FY.F\Delta \rightarrow F(X.Y).F\Delta \rightarrow F(X.Y.\Delta) \rightarrow F(X.\Delta.Y.\Delta),$$
which holds because of the naturality and coherence of the lax structure of $F$. The second diagram commutes because of a similar argument.

Now, if the colax monoidality morphisms are invertible, then $|-|$ will become strong monoidal, and the original diagrams in the question will commute. Hence $\tau$ will become a monoidal transformation. This is the case in the simplicial-topological example.

For the purpose of defining a lax structure on $|-|$ we could again look at our examples. In these examples the inverses of the lax structure of $|-|$ are defined as follows. In the topological case the lax monoidal structure $|X|\times|Y| \rightarrow |X\times Y|$ is constructed (-> Theorem 11.5 "The geometry of iterated loops") using certain maps

$$X_n\times X_m \times \Delta^n\times \Delta^m \rightarrow X_{n+m}\times Y_{n+m} \times \Delta^{n+m}$$

which depend on certain isomorphisms $\Delta^n\times\Delta^m \cong \Delta^{n+m}$. In the bisimplicial situation $|X|\times|Y| \rightarrow |X\times Y|$ is explicitly given via

$$(X_n\times Y_m \times \Delta^n\times \Delta^m)_r \rightarrow (X_r\times Y_r \times \Delta^r)_r$$
$$(x, y, u : r \rightarrow n, v : r \rightarrow m) \mapsto (u^\ast(x), v^\ast(y), 1 : r \rightarrow r).$$

However, these do not have straightforward generalization to the monoidal context.