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We consider the neutrino self-energy in a background composed of a scalar particle and a fermion using a simple model for the coupling of the form λf¯RνLϕ. The results are useful in the context of dark matter-neutrino interaction models in which the scalar and/or fermion constitute the dark matter. The corresponding formulas for models in which the scalar particle couples to two neutrinos via a coupling of the form λ(ννϕ)ν¯RcνLϕ are then obtained as a special case. In the presence of these interactions there can be new contributions to the neutrino effective potential and index of refraction in the context of neutrino collective oscillations in a supernova and in the early Universe hot plasma before neutrino decoupling. The formulas obtained here can be used to estimate those effects and/or put limits on the model parameters based on the contribution that such interactions can have in those contexts. A particular feature of the results is that the contribution to the neutrino self-energy or effective potential in a neutrino background due to the ννϕ coupling is proportional to the antineutrino-neutrino asymmetry (nν¯−nν), in contrast to the standard Z contribution which is proportional to (nν−nν¯). Therefore the two contributions tend to cancel. If the cancellation is significant, it is conceivable that the O(1/mϕ4) terms give the dominant contribution. Alternatively, a limit can be set by requiring that the contribution of the ννϕ interaction to the neutrino effective potential does not cancel the standard contribution in an appreciable way.