  Inelastic neutron scattering is used to study the magnetic excitations of the
$S=1$ square-lattice antiferromagnet La$_2$NiO$_4$. We find that the spin waves
cannot be described by a simple classical (harmonic) Heisenberg model with only
nearest-neighbor interactions. The spin-wave dispersion measured along the
antiferromagnetic Brillouin-zone boundary shows a minimum energy at the
$(1/2,0)$ position as is observed in some $S=1/2$ square-lattice
antiferromagnets. Thus, our results suggest that the quantum dispersion
renormalization effects or longer-range exchange interactions observed in
cuprates and other $S=1/2$ square-lattice antiferromagnets are also present in
La$_2$NiO$_4$. We also find that the overall intensity of the spin-wave
excitations is suppressed relative to linear spin-wave theory indicating that
covalency is important. Two-magnon scattering is also observed.
