We study the effect of nonmagnetic Zn²⁺ (spin-0) and magnetic Ni²⁺ (spin-1) impurities on the ground state and low-lying excitations of the quasi-one-dimensional spin-1/2 Heisenberg antiferromagnet Sr₂CuO₃ using inelastic neutron scattering, specific heat, and bulk magnetization measurements. We show that 1% Ni²⁺ doping in Sr₂CuO₃ results in a sizable spin gap in the spinon excitations, analogous to the case of Ni doped SrCuO₂ previously reported [Simutis et al., Phys. Rev. Lett. 111, 067204 (2013)]. However, a similar level of Zn²⁺ doping in SrCuO₂, investigated here for comparison, did not reveal any signs of a spin gap. Magnetic ordering temperature was found to be suppressed in the presence of both Zn²⁺ and Ni²⁺ impurities; however, the rate of suppression due to Ni²⁺ was found to be much more pronounced than for Zn²⁺. Effect of magnetic field on the ordering temperature is investigated. We found that with increasing magnetic field, not only the magnetic ordering temperature gradually increases but the size of specific heat anomaly associated with the magnetic ordering also progressively enhances, which can be qualitatively understood as due to the field induced suppression of quantum fluctuations.