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There is a simple example in the case of the Hartman-Grobman theorem for maps in 3D. The example appears in the original article paper 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 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", Bol. Soc. Mat. Mexicana (2) 5, 1960).

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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 $C^{1}$.
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