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).
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 $C^{1}$.