You need essentially the same condition as in the case of the domain $x\in\mathbb R$. That is, $$u(x,t)=o(e^{\epsilon|x|^2})$$
for every $\epsilon>0$.

**Edit**. Tikhonov provided an example of a non-trivial solution of the heat equation on the domain $\mathbb R$, with zero data. Take either its odd part, or the derivative of its even part with respect to $x$. It is a non-trivial solution of the heat equation in the domain $(0,+\infty)$ with zero Dirichlet boundary condition and zero initial data. If such a principle as the one considered by the MO author existed, this solution would be trivial.