I have found only one reason, this being polarisation isn't accounted for. See for example this PDF - it is in the middle of the second paragraph, first page.
What other reasons lead to the failure of Lennard-Jones potential in simple molecules?
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The Lennard-Jones potential arises from the mutual polarisation of the two molecules. It is present for all molecules however it is a weak interaction and easily overwhelmed by electromagnetic forces. For example two ions would interact via their electric fields much more strongly than via the LJ mechanism and we'd get a $1/r^2$ force ($1/r$ potential).
Ammonia is neutral, but it has a large electric dipole moment and this dominates the interaction. Dipolar forces fall off as rough $1/r^3$, but they are also strongly dependent on the mutual orientation of the two molecules so the potential is not isotropic. The LJ potential will still be present and make a contribution to the total potential, but it will be very small compared to the dipole-dipole interaction.