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"][1],  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 $C^{1}$. 


  [1]: http://www.ams.org/mathscinet/search/publdoc.html?arg3=&co4=AND&co5=AND&co6=AND&co7=AND&dr=all&pg4=AUCN&pg5=TI&pg6=PC&pg7=ALLF&pg8=ET&r=1&review_format=html&s4=Hartman&s5=%2520structural%2520stability&s6=&s7=&s8=All&vfpref=html&yearRangeFirst=&yearRangeSecond=&yrop=eq