Let $X$ be a space and $\mathcal U$ a cover of $X$. Instead of Čech cohomology, I would like to take the following construction:
let
$$I= \{ \text{finite nonempty intersections of elements of }\,\mathcal U\},$$
which is a poset, and now I can take the nerve $N(I)$ in the sense of category theory,
whose $n$-simplices are chains of length $n+1$ in $N(I)$.
Given a sheaf $\mathcal F$ on $X$, I get a local system of coefficients $F$ on $N(I)$ by taking $F(\sigma) = \mathcal F(\min \sigma)$.
I would like to relate $H^\*(N(I), F)$ to the sheaf cohomology $H^\*(X,\mathcal F)$. Do you know how to do this?


My hope is that this is sheaf cohomology if we assume enough acyclicity about $\mathcal F$ and $I$. This complex smells similar to the Čech complex, but I am not sure of the general relation. If $X$ was a simplicial complex and $\mathcal U$ the covering by star neighborhoods, then we recover Čech cohomology with respect to star neighborhoods in the barycentric subdivision.


In my situation $X$ is a quasi-compact separated scheme, $\mathcal U$ is a cover by affine opens, and $\mathcal F$ is a quasi-coherent sheaf. But, the same setup might work if we just assume $\mathcal F$ is acyclic on every element of $I$.



Čech-like cohomology with the “other nerve”
This construction as stated gives exactly the barycentric subdivision of the Čech nerve of $\mathcal{V}$, as any simplicial complex yields a face poset and the categorical nerve of that is the barycentric subdivision of the original complex. There are several alternative nerve constructions that can be used; see H. Abels and S. Holz, Higher generation by subgroups, J. Alg, 160, (1993), 311– 341, for a discussion on several of them. In your fairly classical case there is also the Vietoris nerve which was explored by Dowker in Homology Groups of Relations, Annals of Maths, 56, (1952), 84 – 95.


