We study tunneling ionization in a one-dimensional three-body model of a molecule, treating both electronic and nuclear degrees of freedom exactly. In a recent paper [Phys. Rev. Lett. 111, 153003 (2013)] it was demonstrated using a finite-range potential that the Born-Oppenheimer approximation breaks down for fields weaker than a critical field FBO when describing tunneling ionization of molecules, but works for stronger fields. It was also demonstrated that the weak-field asymptotic theory allows for an accurate description in this weak-field limit. In the present paper, we consider a potential with a Coulomb tail and nonzero dipole moment, modeling polar molecules. Our study shows that the conclusions of the aforementioned paper also hold for this potential.