We report a Si-29 NMR study on the pressure-induced superconductivity (SC) in an antiferromagnetic (AFM) heavy-fermion compound CeIrSi3 without inversion symmetry. In the SC state at P=2.7-2.8 GPa, the temperature (T) dependence of the nuclear-spin lattice relaxation rate 1/T-1 below T-c exhibits a T-3 behavior without any coherence peak just below T-c, revealing the presence of line nodes in the SC gap. In the normal state, 1/T-1 follows a root T-like behavior, suggesting that the SC emerges under the non-Fermi-liquid state dominated by AFM spin fluctuations enhanced around a quantum critical point. The reason why the maximum T-c in CeIrSi3 is relatively high among the Ce-based heavy-fermion superconductors may be the existence of the strong AFM spin fluctuations. We discuss the comparison with the other Ce-based heavy-fermion superconductors.