If $\rho : K \rightarrow X$ is a universal cover (basepointed) are there simple necessary and sufficient conditons for $\rho^* :\langle X, C \rangle \rightarrow \langle K, C \rangle $ for any CW complex $C$ to be trivial?
For example, since $\mathbb{R}$ is contractible, the covering of the circle has this property. However, this is not always the case since for any $K(G,n)$ with $n>1$ and $G$ non trivial we know that the identity is a universal covering that doesn't have $\rho ^*$ trivial since we can think of $\rho^* $ as the identity map on $ H^n (K(G,n),G)$ which is not trivial.
(As your comment suggests, i'm assuming $K,X$ are CW-complexes too)
I think your comment settles it : putting $C=X$ and taking $id_X$ you get that $\rho$ is nullhomotopic.
In particular, $\rho$ induces $0$ on all homotopy groups. But $\rho$ induces isomorphisms on higher homotopy groups, so $\pi_n(X)=0$ for $n\geq 2$.
Now this implies that $X=K(\pi,1) =B\pi$ for $\pi=\pi_1(X)$; but then we are in the situation you described with $\mathbb{R}$ : $K$ is the universal cover of an aspherical space, so it is contractible, and so $\langle K,C\rangle$ is always trivial anyway.
So this happens if and only if $X$ is a $K(\pi,1)$