I can offer some hints:
what do you see in a movie of a trajectory? Is it the dude particles being pulled away from the cores? something else?
your system contains only a few ions. Is the volume sufficient to avoid minimum image issues?
which atom type is Li+? Is it 1? The polarisability of Li+ would be needed to setup a drude dipole on Li+. But it should be very small, so a non-polarisable cation should be ok (as for H).
what is the force field for the ionic liquid? I am surprised not to see
special_bonds lj/coul 0.0 0.0 0.5 extra 1
and also to see only one type of H (14). I’d expect the H close to the positive N to have different charges.