Consider the category of pointed simplicial sets with usual notion of weak equivalence. The question is does the functor 
$$Y \mapsto X\wedge Y$$ preserve weak equivalences? or at the very least does it preserve trivial cofibrations (monomorphisms+weak equivalences)?



smash product of pointed spaces preserve weak equivalences
Yes, $\wedge$ is invariant under weak homotopy equivalence in $\mathbf{sSet}\_\*$. It suffices to show that it sends anodyne extensions (= trivial cofibrations) to weak homtoopy equivalences: indeed, every pointed simplicial set is cofibrant, so by Ken Brown's lemma, any endofunctor on $\mathbf{sSet}\_\*$ that sends anodyne extensions (= trivial cofibrations) to weak homotopy equivalences must preserve all weak homotopy equivalences.


Let $X \vee Y = X \times \{ \* \} \cup \{ \* \} \times Y \subseteq X \times Y$. It is easy to see that $X \vee (-)$ and $(-) \vee Y$ each preserve anodyne extensions, so $\vee$ is invariant under weak homotopy equivalence. Recall that $X \wedge Y$ is defined by the following pushout diagram in $\mathbf{sSet}\_\*$:
$$\require{AMScd}
\begin{CD}
X \vee Y @>>> X \times Y \\
@VVV @VVV \\
\Delta^0 @>>> X \wedge Y
\end{CD}$$
Suppose $X\_0 \to X\_1$ and $Y\_0 \to Y\_1$ are anodyne extensions. Consider the following diagram in $\mathbf{sSet}\_\*$,
$$\begin{CD}
X\_0 \vee Y\_0 @>>> X\_1 \vee Y\_1 @>>> X\_1 \times Y\_1 \\
@VVV @VVV @VVV \\
\Delta^0 @>>> Z\_0 @>>> Z\_1 \\
& @VVV @VVV \\
&& \Delta^0 @>>> X\_1 \wedge Y\_1
\end{CD}$$
where every square is a pushout square. Since $X\_0 \vee Y\_0 \to X\_1 \vee Y\_1$ is an anodyne extension, the same is true of $\Delta^0 \to Z\_0$, so $Z\_0 \to \Delta^0$ is a weak homotopy equivalence. Moreover, $X\_1 \vee Y\_1 \to X\_1 \times Y\_1$ is a monomorphism (= cofibration), so $Z\_0 \to Z\_1$ is also a monomorphism. Since the model structure on $\mathbf{sSet}\_\*$ is left proper, it follows that $Z\_1 \to X\_1 \vee Y\_1$ is a weak homotopy equivalence.


On the other hand, we also have the following diagram in $\mathbf{sSet}\_\*$,
$$\begin{CD}
X\_0 \vee Y\_0 @>>> X\_0 \times Y\_0 @>>> X\_1 \times Y\_1 \\
@VVV @VVV @VVV \\
\Delta^0 @>>> X\_0 \wedge Y\_0 @>>> Z\_1
\end{CD}$$
where both squares are pushout squares, and $X\_0 \times Y\_0 \to X\_1 \times Y\_1$ is an anodyne extension (because the model structure on $\mathbf{sSet}$ is cartesian), so $X\_0 \wedge Y\_0 \to Z\_1$ is also an anodyne extension. It is easy to check that the composite $X\_0 \wedge Y\_0 \to Z\_1 \to X\_1 \wedge Y\_1$ is the morphism obtained by functoriality, so this completes the proof.


