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 $n\ge 2$ there exists a representation $\rho: \pi_1(X)\to SO(n)$ with vanishing homology.