Formulation of the Gauss-Ostrogradski theorem from my mathematical analysis notes is as such:
Let $N\in\mathbb{N},\,N\ge2$ and $\Omega\subset\mathbb{R}^N$ is an open and bounded set with its boundary being a generalized $(N-1)$ surface, such that its surface area, $S_{N-1}(\partial\Omega)<\infty$. Let there exist an external normal vector $\vec{\nu}(\vec{x})$ at every point $\vec{x}$ of each of the $\partial\Omega$'s $(N-1)$ dimensional component. Also, let $i\in\{1,\ldots,N\}$ and $F\in C(\bar{\Omega})$, such that $\frac{\partial F}{\partial x_i}$ exists everywhere in $\Omega$ and can be continuously extended onto $\bar{\Omega}$. Then:
$$\int_\Omega \frac{\partial F}{\partial x_i}\,\mathrm{d}x =\int_{\partial\Omega} F\nu_i\,\mathrm{d}S\tag{1}$$
What happens if the surface area is infinite?
Such integrals are encountered in the Boltzmann velocity distribution function context and the G-O theorem is used often but I have not found anywhere the statement with infinite surface area, one usually has:
$$\int \frac{\partial }{\partial v_i}\left(\varphi(t,\vec{r},\vec{v}) f(t,\vec{r},\vec{v})\right)\,\mathrm{d}\vec{v}\tag{2}$$ where the integration is over the whole velocity space, which (I assume) is just $\mathbb{R}^3$.
Would it be sufficient for the G-O theorem proof to have such $F$ in (1) that the integral on the right hand side above is finite in a limit over a sphere: $$\lim_{r\to\infty}\int_{\partial B_r} F(\vec{x})\nu_i\mathrm{d}S<\infty\tag{3}$$
And what would be the necessary conditions on the function $F$?
The assumption that $\Omega$ is bounded implies that $\partial \Omega$ is bounded. Since $\partial \Omega$ is also closed, this means it is compact and therefore has finite surface area.
However, the theorem remains true for $C^1$ boundary open sets $\Omega$ that are unbounded, provided that $F$ has compact support. This is because if $F$ has compact support and is smooth on $\mathbb{R}^n$, you can use a partition of unity to localize the analysis and then prove the theorem directly. Of course, there are further ways you can weaken hypotheses, involving Lipschitz functions and Sobolev functions.