What is the meaning of a non-resonant neutrino production?

The resonance they are talking about is very much the same kind of resonance that you quote from Wikipedia. If you look at equation (6.6) of reference [1] that you gave, you see that the evolution of the sterile neutrino phase space density $f_s$ is governed by this big fraction on the right hand side. In the denominator you find the potentials $V_L$ and $V_T$ both with a negative sign. If large enough, they might compensate the other terms in the denominator to a very high degree. This would make the denominator very small and thus the right hand side of equation (6.6), which one can think of as the sterile neutrino production rate, very big. This is what is meant by resonance. As it turns out, for a resonance to occur one needs the potential $V_L$, which in return requires a primordial lepton asymmetry. If you don't have such an asymmetry, there will be no resonance and the sterile neutrino production will take place only non-resonantly, which is what the authors of [1] say.