Does the classifying space of monoids commute with wedge sum up to weak equivalence? If $G$ and $H$ are groups, then the map $BG\vee BH\to B(G\ast H)$ is a weak equivalence by van Kampen's theorem. However the classifying space $BM$ of a monoid can have arbitrary homotopy type so higher homotopy groups are involved.
If $M$ and $N$ are monoids, is the map $BM\vee BN\to B(M\ast N)$ still a weak equivalence? If there are counterexamples, is there still some criterion that insures such a weak equivalence?
 A: The answer is yes.  This is a special case of Theorem 4.1 of http://www.jstor.org/stable/10.2307/2374307 which gives a sufficient condition for a pushout diagram of monoids to be transformed into a pushout diagram of classifying spaces.
For those without access, the result is that if the $M_i$ are a family of monoids with a common submonoid $W$ such that $\mathbb ZM_i$ is flat over $\mathbb ZW$ for all $i$, then the pushout of the $BM_i$ over $BW$ is a classifying space for the pushout out of the $M_i$ over $W$.  Clearly, if $W$ is trivial (the free product case), the condition holds.
Notice the flatness condition is free for groups (pun intended) since the group algebra is free over the group algebra of a subgroup.  Also the proof is very close to the one I tried in my previous (now deleted) answer.
A: Charles has given a very good answer to the question. 
The following is not meant to be an answer, but just a heuristic argument which I cannot 
make into a proof. 
There should be an operation, "free product," denoted $\sharp$, in the category of associative topological monoids. If $X,Y$ are based spaces, then 
$$\Omega (X \vee Y)$$ (Moore loops), should decompose (at least up to homotopy) in this category as
$$(\Omega X) \sharp (\Omega Y) .$$ The reason  I find this to be plausible is that a loop in $X \vee Y$ is clearly a word of loops of $X$ and $Y$, and a word is supposed to represent an element of the free product.
Supposing this to be the case, we could take $X = BM$ and $Y = BN$, then we would have
$$
\Omega (BM \vee BN) \simeq (\Omega BM) \sharp (\Omega BN)
$$
It should also be the case that the inclusion 
$$M \ast N \to (\Omega BM) \sharp (\Omega BN)$$ 
is group completion, since $ (\Omega BM)$ and $ (\Omega BN)$ are group-like and the operation $\sharp$ should preserve grouplike monoids (furtheremore, we also should have
$M \ast N \simeq M\sharp N$). If this is true, then $(\Omega BM) \sharp (\Omega BN) \simeq \Omega B (M\ast N)$.
If the above works, then the homomorphism
$$
\Omega (BM \vee BN) \to (\Omega BM) \sharp (\Omega BN)
$$
is an equivalence. Now apply the classifying space to get the desired equivalence
$BM \vee BN \simeq B(\Omega BM) \sharp (\Omega BN) \simeq B(M \ast N)$.
Question: Can this heuristic sketch be made into a proof?
A: Here is a high-tech point of view.
The inclusion functor $\mathrm{Groups}\to \mathrm{Monoids}$, has a left adjoint $F\colon \mathrm{Monoids}\to\mathrm{Groups}$, which is the group completion functor.  You know that 
$$\pi_1(BM) = FM.$$
The claim is that if we instead consider the total left derived functor $LF$ of $F$ ("derived group completion"), then we get a sharper result, namely
$$BM \approx B(LF(M))$$
where $\approx$ is weak equivalence.  This should apply for any simplicial monoid $M$ (or topological monoid, if you prefer).  
Being a left derived functor, $LF$ must commute with homotopy colimits.  If you also know that:


*

*The free product of any two discrete monoids is weakly equivalent to their homotopy coproduct as simplicial monoids, and 

*the homotopy theory of simplicial groups is equivalent to the homotopy theory of pointed connected spaces,
then the result follows.  
I think the paper "Simplicial localizations of categories" by Dwyer and Kan (http://www.ams.org/mathscinet-getitem?mr=579087) has a nice treatment of these kinds of ideas.  (They actually discuss the derived functor of the construction $(C,W)\mapsto C[W^{-1}]$, which associates a category of fractions to a category with a distinguished subcategory.  In the case where $W=C$, this is groupoid completion; they show that the classifying space of the derived groupoid completion of $C$ is weakly equivalent to the nerve of $C$.)
Added:  It appears that the statement you want is proven (in simplicial language) as Proposition 3.8 of the Dwyer-Kan paper I linked to.  In that paper, they work with categories which all have the same object set $O$; when $O$ is a singleton, the lemma exactly says that $N(D*E)\approx N(D)\vee N(E)$, where $N$ is the nerve.
