  $^{31}$P nuclear magnetic resonance (NMR) measurements have been carried out
to investigate the magnetic properties and spin dynamics of Fe$^{3+}$ ($S$ =
5/2) spins in the two-dimensional triangular lattice (TL) compound
Na$_3$Fe(PO$_4$)$_2$. The temperature ($T$) dependence of nuclear spin-lattice
relaxation rates ($1/T_1$) shows a clear peak around N\'eel temperature,
$T_{\rm N} = 10.9$~K, corresponding to an antiferromagnetic (AFM) transition.
From the temparature dependence of NMR shift ($K$) above $T_{\rm N}$, an
exchange coupling between Fe$^{3+}$ spins was estimated to be $J/k_{\rm
B}\simeq 1.9$~K using the spin-5/2 Heisenberg isotropic-TL model. The
temperature dependence of $1/T_1T$ divided by the magnetic susceptibility
($\chi$), $1/T_1T\chi$, above $T_{\rm N}$ proves the AFM nature of spin
fluctuations below $\sim$ 50 K in the paramagnetic state. In the magnetically
ordered state below $T_{\rm N}$, the characteristic rectangular shape of the
NMR spectra is observed, indicative of a commensurate AFM state in its ground
state. The strong temperature dependence of 1/$T_1$ in the AFM state is well
explained by the two-magnon (Raman) process of the spin waves in a 3D
antiferromagnet with a spin-anisotropy energy gap of 5.7 K. The temperature
dependence of sublattice magnetization is also well reproduced by the spin
waves. Those results indicate that the magnetically ordered state of
Na$_3$Fe(PO$_4$)$_2$ is a conventional 3D AFM state, and no obvious spin
frustration effects were detected in its ground state.
