Does the nerve functor (resp. fundamental groupoid functor) preserve homotopy colimits (resp. homotopy limits)? Let $\pi _1:SS\to Grpd$ denote the fundamental groupoid functor, from simplicial sets to groupoids, and let $N:Grpd\to SS$ denote the nerve functor. Then $\pi _1$ is left adjoint to $N.$
On simplicial sets we consider the usual model category and on groupoids the model category structure inherited by the one given by Thomason. It is known that $\pi _1$ preserves weak equivalences, cofibrations, fibrations and colimits, and that $N$ preserves weak equivalences, fibrations and limits. From these facts one can easily deduce that $\pi _1$ preserves homotopy colimits and $N$ preserves homotopy limits. I wonder whether the following questions have positive answers:


*

*Does $N$ preserve homotopy colimits?

*Does $\pi _1$ preserve homotopy limits?

 A: The fundamental groupoid does not preserve homotopy limits. The simplest examples can be obtained from the path-space sequence $\Omega X\to \operatorname{pt}\to X$ which shows that $\Omega X$ is the homotopy limit of $\operatorname{pt}\to X\leftarrow \operatorname{pt}$. If $\pi_1$ preserves homotopy limits, then the fundamental groupoid of $\Omega X$ should be contractible for any simply-connected space $X$. But any abelian group can be realized as $\pi_2(X)$ for $X$ simply-connected, so the fundamental groupoid can not preserve homotopy limits. 
A: Just for fun, here's a purely abstract way of seeing neither of these can happen:
The functors you wrote down both preserve weak equivalences, so are morphisms of relative categories, and induce adjunctions of the associated $\left(\infty,1\right)$-categories, which are $\infty$-groupoids and $1$-groupoids (which is actually a $(2,1)$-category) respectively. The adjunction is precisely the one exhibiting $1$-groupoids as a reflective subcategory of $\infty$-groupoids. The full and faithful inclusion is modeled by $N$. The $\left(\infty,1\right)$-category of $\infty$-groupoids is freely generated by the (any) contractible $\infty$-groupoid (i.e. the one point space), (i.e. it's the free colimit cocompletion of the terminal category). Since the $\left(2,1\right)$-category of groupoids is cocomplete, and contains the terminal groupoid, if the inclusion preserved colimits, it would have to be essentially surjective:
If $X$ is any $\infty$-groupoid, you can write $X$ as the colimit of the terminal functor $X \to \infty\mbox{-}\mathbf{Gpd}$ (the constant functor with value the point) (this is basically just the Yoneda Lemma for $\infty$-categories), and this functor (obviously) factors through the inclusion $\mathbf{Gpd} \hookrightarrow \infty\mbox{-}\mathbf{Gpd}$.
If the left adjoint $\tau_1$ of this inclusion (which is modeled by $\Pi_1$) preserved even finite limits, then this would exhibit the $(2,1)$-category of groupoids as a left-exact localization of $\infty\mbox{-}\mathbf{Gpd}=\mathbf{Psh}_\infty\left(*\right),$ i.e. it would exhibit $\mathbf{Gpd}$ as an $\infty$-topos. This means there would exist a unique Grothendieck topology $J$ on the terminal category such that $\mathbf{Gpd}$ sat somewhere between $\mathbf{Sh}_\infty\left(*,J\right)$ and its hypercompletion $\mathbf{HSh}_\infty\left(*,J\right).$ But, there is a unique Grothendieck topology on the terminal category (the trivial one!), so we'd have $$\mathbf{HSh}_\infty\left(*,J\right)=\mathbf{Sh}_\infty\left(*,J\right)=\mathbf{Psh}_\infty\left(*\right).$$ So we'd have to have $\mathbf{Gpd}\simeq \mathbf{Psh}_\infty\left(*\right)=\infty\mbox{-}\mathbf{Gpd}$ which is clearly nonsense. 
A: The nerve functor does not preserve homotopy colimits. Indeed, take any simplicial set $X$ with non-trivial $\pi_n$ ($n > 1$) and consider $X$ as a simplicial diagram of sets. In $\mathbf{sSet}$, its homotopy colimit will be $X$ (up to weak equivalence, by the Bousfield–Kan theorem), but the nerve of any 1-groupoid has trivial $\pi_n$ ($n > 1$).
The fundamental groupoid functor does not preserve homotopy limits. As Matthias Wendt explained, the loop space can be computed by a homotopy limit, so if we choose a simplicial set $X$ with trivial $\pi_0$ and $\pi_1$ but non-trivial $\pi_2$, then the homotopy pullback diagram
$$\begin{array}{ccc}
\Omega X & \rightarrow & \Delta^0 \\
\downarrow & & \downarrow \\
\Delta^0 & \rightarrow & X
\end{array}$$
cannot be preserved by the fundamental groupoid functor.
