There is a simple example in the case of the Hartman-Grobman theorem for *maps* in 3D. The example appears in the original paper 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 any linearizing map, then both $\varphi$ and $\varphi^{-1}$ are **not** of class $C^{1}$. 

In the 2D case, one can show that any map $T(X)=AX+F(X)$ of class $C^2$ such that $F$ and its gradient vanish at $X=0$ can be  linearized in the neighborhood of $X=0$ with a $C^1$-diffeomorphism, provided that the matrix $A$ has no eigenvalue of absolute value of $0$ or $1$ (see another paper by Hartman, ["On local homeomorphisms of Euclidean spaces"][2], 
Bol. Soc. Mat. Mexicana (2)  5,  1960).


  [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
  [2]: 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=on%2520local%2520homeomorphisms&s6=&s7=&s8=All&vfpref=html&yearRangeFirst=&yearRangeSecond=&yrop=eq