For any $m\in\mathbb{N}$ let $\pi_m$ denote the projector on the space $V_m:=\{x\in \ell_\infty\, : \, x_j=0,\,\, \forall j>m\}$. Let $S$ denote the left shift operator on $ \ell_\infty$, that is $Sx:=( x_2,x_3,\dots)$ for any $x:=(x_1,x_2\dots)\in \ell_\infty$. Consider the bounded linear operator (a norm-$2$ projector onto $V_m$, indeed)

$$P_m:=\pi_m(I-S^m/2)^{-1}=\pi_m \sum_{j=0}^\infty 2^{-j}S^{jm}\, .$$ Finally, consider the set $A:=\{0,1\}^\mathbb{N_+}=\{x\in \ell_\infty\, : \forall j\,\, x_j\in\{0,1\} \} $. For any $m$ the set $\pi_m(A)=\{0,1\}^m$ is finite, so $\mathcal{H}^k(\pi_m(A))=0$ for any $k>0$. On the other hand, $P_m(A)=[0,1]^m$, so $1=\mathcal{H}^m(P_m A)\le \|P_m\|^m \mathcal{H}^m(A)$ for any $m$, whence $\mathcal{H}^m(A)=\infty$ for any $m$.

For any $m\in\mathbb{N}$ let $\pi_m$ denote the projector on the space $V_m:=\{x\in \ell_\infty\, : \, x_j=0,\,\, \forall j>m\}$. Let $S$ denote the left shift operator on $ \ell_\infty$, that is $Sx:=( x_2,x_3,\dots)$ for any $x:=(x_1,x_2\dots)\in \ell_\infty$. Consider the bounded linear operator (a norm-$2$ projector onto $V_m$, indeed)

$$P_m:=\pi_m(I-S^m/2)^{-1}=\pi_m \sum_{j=0}^\infty 2^{-j}S^{jm}\, .$$ Finally, consider the set $A:=\{0,1\}^\mathbb{N_+}=\{x\in \ell_\infty\, : \forall j\,\, x_j\in\{0,1\} \} $. For any $m$ the set $\pi_m(A)=\{0,1\}^m$ is finite, so $\mathcal{H}^k(\pi_m(A))=0$ for any $k>0$. On the other hand, $P_m(A)=[0,2]^m$, so $2^m=\mathcal{H}^m(P_m A)\le \|P_m\|^m \mathcal{H}^m(A)$ for any $m$, whence $\mathcal{H}^m(A)\ge 1$ for any $m$, so in fact $\mathcal{H}^m(A)= \infty$ for any $m$.