In a nut-shell: exchange integrals are two-electron integrals, and two-electron integrals yield positive values. Note that the "kind" or "meaning" of the input functions is irrelevant, because in practice, you will always have linear combinations of primitives, and in most cases gaussians. For the proof of the claim about positive values, I will defer to the experts, [1, HJO] who cite previous work. [2] As taken from the book:
The two-electron intergrals can be viewed as a matrix with the electron distributions [($\Omega_{ab}, \Omega_{cd}$)] as row and column labels [using AO labels $a,b,c,d$, see above]
$$
g_{abcd} = \int \int \frac{\Omega_{ab}(\mathbf{r}_1) \Omega_{cd}(\mathbf{r}_2)}{r_{12}} \mathrm{d}\mathbf{r}_1 \mathrm{d}\mathbf{r}_2
$$
Assuming that the orbitals are real, we shall demonstrate that this matrix is positive definite [2]. Let us consider the interaction between two electrons in the same distribution $\rho(\mathbf{r})$:
$$
I[\rho] = \int \int \frac{\rho(\mathbf{r}_1) \rho(\mathbf{r}_2)}{r_{12}} \mathrm{d}\mathbf{r}_1 \mathrm{d}\mathbf{r}_2
$$
Inserting the Fourier transform of the interaction operator
$$
\frac{1}{r_{12}} = \frac{1}{2\pi^{2}} \int k^{-2} \exp[\mathrm{i}\mathbf{k} \cdot(\mathbf{r}_1 - \mathbf{r}_2)] \mathrm{d}\mathbf{k}
$$
and carrying out the integration over the Cartesian coordinates, we obtain
$$
I[\rho] = \frac{1}{2\pi^{2}} \int k^{-2} \vert \rho(\mathbf{k}) \vert^2 \mathrm{d}\mathbf{k} \quad\quad \text{(eq. 4)}
$$
where we have introduced the distributions
$$
\rho(\mathbf{k}) = \int \exp(-\mathrm{i}\mathbf{k}\cdot\mathbf{r}) \rho(\mathbf{r}) \mathrm{d}\mathbf{r}
$$
Since the integrand in [(eq. 4)] is always positive or zero, we obtain the inequality
$$
I[\rho] > 0
$$
HJO go on to expand the charge distribution $\rho$ in one-electron orbital distributions and get back to the original $g_{abcd}$, noting afterwards that two-electrons thus satisfy the conditions for inner products, in a metric defined by $r^{-1}_{12}$. Therefore, Schwarz-style inequalities hold and are used extensively in integral screening to throw out insignificant integrals before evaluating them.
[1] T Helgaker, P Jørgensen, J Olsen, Molecular Electronic-Structure Theory, Wiley (2002), p. 403f.
[2] CCJ Roothaan, Rev. Mod. Phys., 23, 69 (1951).