Let's consider $H_0 = \{X\sim\mathcal N_{\mu, \sigma^2}\}$ and $H_1\{X\sim Poi(\lambda)\}$. I need to find a test $\varphi$ that has type $1$ and type $2$ errors of $0$.
I came up with $$\varphi = \left\lbrace\begin{array}&1 & X \in \mathbb R\setminus \mathbb N \\ 0 & X\in\mathbb N \end{array} \right.$$ because $Poi(\lambda)$ is discrete and $\mathcal N_{\mu, \sigma^2}$ is continuous. Does this work?
Let $T(X)\in\{0,1\}$ be a test on a single sample $X$. Our hypotheses are $H_0:X\sim \mathcal{N},\,H_1: X \sim \textrm{Poisson}$. We reject $H_0$ if $T(X)=1$. Then $E_{X\sim \mathcal{N}}[T(X)]=P_{X\sim \mathcal{N}}(T(X)=1)$ is the Type I error probability and $1-E_{X\sim \textrm{Poisson}}[T(X)]=P_{X\sim \textrm{Poisson}}(T(X)=0)$ is the Type II error probability. Now recall $P_{X\sim \mathcal{N}}(X=n)=0$ for all $n \in \mathbb{N}_0$, while $\sum_{n\in \mathbb{N}_0}P_{X\sim \textrm{Poisson}}(X=n)=1$. Let $T(x)=\mathbf{1}_{\mathbb{N}_0}(x)$. Then $$\begin{aligned}&E_{X\sim \mathcal{N}}[\mathbf{1}_{\mathbb{N}_0}(X)]=P_{X\sim \mathcal{N}}(X\in \mathbb{N}_0)=0\\ &E_{X\sim \textrm{Poisson}}[\mathbf{1}_{\mathbb{N}_0}(X)]=P_{X\sim \textrm{Poisson}}(X\in \mathbb{N}_0)=1 \end{aligned}$$