triviality of homology with local coefficients

Maybe you are looking for something more interesting, but you can take $X=S^1$, universal cover $tilde X$, and $rho: {mathbb Z}to O(n)$ such that the image group has no fixed unit vectors in $R^n$. Then $H_*(tilde X,rho)=0$ (which is a nice exercise to work out if you are new to this material). A more challenging problem would be:

Construct a finite CW-complex $X$ such that for each $nge 2$ there exists a representation $rho: pi_1(X)to SO(n)$ with vanishing homology.

If you are interested in 3-dimensional topology, here are two classes of examples you should be aware of:

a. Suppose that $X$ is a closed connected orientable 3-manifold with finite nontrivial fundamental group $pi$ and $tilde Xto X$ is its universal covering. Then for each $rho: pito O(4)$ such that $rho(pi)$ has no fixed unit vectors, $H_*(tilde X,rho)=0$. (Examples of such $rho$ are given by the fact that $pi$ embeds in $SO(4)$ so that the image group acts freely on $S^3$.)

b. Suppose that $X$ is a closed connected orientable arithmetic hyperbolic 3-manifold and $tilde Xto X$ is its universal covering. Then there exists a representation $rho: pi_1(X)to O(3)$ such that $H_*(tilde X,rho)=0$.

Edit. At the same time, there are spaces for which your property does not hold, for instance a space whose fundamental group admits only trivial orthogonal representations.