We report (23)Na nuclear magnetic resonance (NMR) and zero-field (ZF) and longitudinal-field (LF) muon spin relaxation (μSR) measurements of the depleted hyperkagome compounds Na(4-x)Ir3O8 (x = 0.3 and 0.7), which undergo an insulator-semimetal transition as a function of x. The (23)Na spin-lattice relaxation rates, T1(-1), follow a T(2.5) power law behavior at accessible temperatures of T = 120-350 K. A substantial temperature dependence of T1(-1) indicates the presence of gapped excitations at elevated temperatures through the transition to a semimetallic phase. ZF-μSR results reveal that hole-doping leads to a melting of quasi-static order to a dynamically fluctuating state. The very slow muon depolarization rate which varies hardly with temperature indicates that spins are close to an itinerant limit in the largest doping x = 0.7. The dynamic relaxation rates extracted from the LF-μSR spectra show a three-dimensional diffusive transport. Our combined NMR and μSR results suggest the occurrence of intriguing spin and charge excitations across the insulator-semimetal transition.