  We present a microscopic study of a doped quantum spin liquid candidate, the
hyperkagome Na$_3$Ir$_3$O$_8$ compound by using $^{23}$Na NMR. We determine the
intrinsic behavior of the uniform \textbf{q} $ = 0$ susceptibility via shift
measurements and the dynamical response by probing the spin-lattice relaxation
rate. Throughout the studied temperature range, the susceptibility is
consistent with a semimetal behavior, though with electronic bands
substantially modified by correlations. Remarkably, the antiferromagnetic
fluctuations present in the insulating parent compound Na$_4$Ir$_3$O$_8$
survive in the studied compound. The spin dynamics are consistent with 120$^o$
excitations modes displaying short-range correlations.
