I suspect that the following result should be true and more or less well known:


Let $\mathcal{M}$ be a model category and $I$ a small category with contractible nerve. For every diagram $X: I \to \mathcal{M}$ and every $i \in I$, the natural map $holim\_I X \to X(i)$ is a weak equivalence.


Indeed this is a direct generalization of 9.10 in [D.M. Kan et al. A classification theorem for diagrams of simplicial sets], where the result is stated for $\mathcal{M}$ the model category of simplicial sets. Has the general result been written up? I couldn't find anything in Hirschhorn or Hovey.



Homotopy limits of homotopically constant diagrams over contractible categories
It is true. You can reduce to the case of simplicial sets by using the fact that $\mathbf{R} \mathrm{Hom} (T, -)$ preserves homotopy limits and (allowing $T$ to vary) is jointly conservative. Finding where this is written down is another matter altogether...


I'll just sketch something more concrete for the case of a simplicial model category $\mathcal{M}$. Let $X : \mathcal{I} \to \mathcal{M}$ be any diagram such that each $X i$ is fibrant. Using the cobar construction of homotopy limits (see e.g. Chapter 18 in [Hirschhorn]), it is immediate that
$$\underline{\mathcal{M}} (T, \operatorname{holim} X) \cong \operatorname{holim} \underline{\mathcal{M}} (T, X)$$
for all cofibrant $T$ in $\mathcal{M}$. On the other hand, $\underline{\mathcal{M}} (T, Z)$ is a model of $\mathbf{R} \mathrm{Hom} (T, Z)$ when $T$ is cofibrant and $Z$ is fibrant. Thus, $\operatorname{holim} X \to X i$ is a weak equivalence in $\mathcal{M}$ if and only if 
$$\operatorname{holim} \underline{\mathcal{M}} (T, X) \to \underline{\mathcal{M}} (T, X i)$$
is a weak homotopy equivalence for every cofibrant object $T$ in $\mathcal{M}$. Hence, the claim reduces to the case where $\mathcal{M} = \mathbf{sSet}$.


The same strategy can be used for many other questions about homotopy (co)limits.


