If
$$ \begin{cases}\partial_{t} u-\Delta u=0, & (t, x) \in[0, T] \times \mathbb{R}^{n} \\ u(0, x)=0, & x \in \mathbb{R}^{n}\end{cases} $$
and for all $(t, x) \in[0, T] \times \mathbb{R}^{n}$ we have $u(t, x) \geq 0$, then $u \equiv 0$.
In my thought, we should impose restriction solution in $B_R$, a bounded ball, and use something like maximal pricipal and let $R\to\infty$. And if a can show that $u(t,x)$'s value on the boundary of $B_R$ decline to zero, then by maximum principle we can get zero solution.