There is a simple example in the case of the Hartman-Grobman theorem for maps in 3D. The example appears in the original article by Hartman, "A lemma in the theory of structural stability of differential equations", Proc. Amer. Math. Soc. 11, 1960.
Let's consider the map $T: \mathbb R^3\to \mathbb R^3$ given by $$ T(x,y,z)=(ax,\ ac(y+b xz),\ cz)),$$ where $a>1$, $b>0$, $0 < c<1$, $ac>1$. If $\varphi$ is a linearizing map, then both $\varphi$ and $\varphi^{-1}$ are not of class $C^{1}$.