  $^7$Li nuclear magnetic resonance (NMR) and terahertz (THz) spectroscopies
are used to probe magnetic excitations and their field dependence in the
hyperhoneycomb Kitaev magnet $\beta$-Li$_2$IrO$_3$. Spin-lattice relaxation
rate ($1/T_1$) measured down to 100\,mK indicates gapless nature of the
excitations at low fields (below $H_c\simeq 2.8$\,T), in contrast to the gapped
magnon excitations found in the honeycomb Kitaev magnet $\alpha$-RuCl$_3$ at
zero applied magnetic field. At higher temperatures in $\beta$-Li$_2$IrO$_3$,
$1/T_1$ passes through a broad maximum without any clear anomaly at the N\'eel
temperature $T_N\simeq 38$\,K, suggesting the abundance of low-energy
excitations that are indeed observed as two peaks in the THz spectra, both
correspond to zone-center magnon excitations. At higher fields (above $H_c$),
an excitation gap opens, and a re-distribution of the THz spectral weight is
observed without any indication of an excitation continuum, in contrast to
$\alpha$-RuCl$_3$ where an excitation continuum was reported.
