I think that the problem encountered in this question is not specificly for DFT. But more in general for QM methods. You say that $E(r\rightarrow \infty) \neq0$. Now let us consider the molecular Hamiltonian:
$$\hat{H}=\hat{T}_N+\hat{T}_e+\hat{U}_{NN}+\hat{U}_{Ne}+\hat{U}_{ee}$$
Here:
The kinetic energy of the nucleis:
$$\hat{T}_N=-\sum_i\frac{\hbar^2}{2M_i}\nabla^2_{R_i}$$
The kinetic energy of the electrons:
$$\hat{T}_e=-\sum_i\frac{\hbar^2}{2m_e}\nabla^2_{r_i}$$
The potential energy between the nucleis:
$$\hat{U}_{NN}=\sum_i\sum_{i>j}\frac{Z_iZ_je^2}{4\pi\epsilon_0|R_i-R_j|}$$
The potential energy between the electrons:
$$\hat{U}_{ee}=\sum_i\sum_{i>j}\frac{e^2}{4\pi\epsilon_0|r_i-r_j|}$$
The potential energy between the nucleis and electrons:
$$\hat{U}_{Ne}=-\sum_i\sum_{j}\frac{Z_ie^2}{4\pi\epsilon_0|R_i-r_j|}$$
We would usually make a calculation under the clamped nuclei approximation, thus we will let the term for the kinetic energy of the nuclei fall out, $\hat{T}_N=0$. Now if we consider the Hamiltonian being an operator depending on the distance between two nucleis. We can write it as:
$$\hat{H}(|R_1-R_2|)=-\sum_i\frac{\hbar^2}{2m_e}\nabla^2_{r_i}+\frac{Z_1Z_2e^2}{4\pi\epsilon_0|R_1-R_2|}+\sum_i\sum_{i>j}\frac{e^2}{4\pi\epsilon_0|r_i-r_j|}+\left[ -\sum_{j}\frac{Z_1e^2}{4\pi\epsilon_0|R_1-r_j|} -\sum_{j}\frac{Z_2e^2}{4\pi\epsilon_0|R_2-r_j|} \right]$$
Now we can consider the limit of $|R_1-R_2|\rightarrow \infty$. It can be seen in the above equation that only one term depends directly on $|R_1-R_2|$, i.e.:
$$\lim_{|R_1-R_2|\rightarrow \infty}\frac{Z_1Z_2e^2}{4\pi\epsilon_0|R_1-R_2|}=0$$
Therfore:
$$\hat{H}(|R_1-R_2|\rightarrow \infty)=-\sum_i\frac{\hbar^2}{2m_e}\nabla^2_{r_i}+\sum_i\sum_{i>j}\frac{e^2}{4\pi\epsilon_0|r_i-r_j|}+\left[ -\sum_{j}\frac{Z_1e^2}{4\pi\epsilon_0|R_1-r_j|} -\sum_{j}\frac{Z_2e^2}{4\pi\epsilon_0|R_2-r_j|} \right]$$
It can be seen that $\hat{H}(|R_1-R_2|\rightarrow \infty) \neq 0$. It have to be noted in the above I have made no assumptions regarding the electrons placements to the nucleis. The terms that depends on the electrons will not be constant when the distance between the nucleis are increased, but will not go to zero. I did not look more explicit at these terms, to conserve simplicity of the argument.
The expectation value will thus not go to zero for infinitly seprated atoms. As a fruit for thought, what is the energy of a single atom? What should then the energy of two atoms at infinite distance be?