Consider a random variable $X$ withthat has a probability density function (PDF) $f(x)$. $X$ being non-negative means that $f(x) = 0$ for $x < 0$. The expectation of $X$ is thus
\begin{equation} \int_{-\infty}^\infty x f(x) \mathrm{d}x = \int_0^\infty x f(x) \mathrm{d}x. \end{equation}
Since $f(x) \geq 0$ for it to be a valid PDF, it follows that
\begin{equation} \int_0^\infty x f(x) \mathrm{d}x = 0 \qquad \Leftrightarrow \qquad f(x) = \delta(x), \end{equation}
where $\delta(x)$ is the Dirac delta function. I.e. for $X$ to have a zero-mean, its PDF needs to be a point mass as zero.