We report the La139 nuclear magnetic resonance (NMR), electron spin resonance (ESR), and complementary magnetometry results on the Ir-based double perovskite compounds La2BIrO6 with strongly magnetic Co2+ (3d7, S=3/2) and nonmagnetic Zn2+ (3d10, S=0) ions at the B site. ESR signals from Ir4+ ions in La2ZnIrO6 and a collective ESR mode of coupled Co2+ spins S=3/2 and Ir4+ pseudospins jeff=1/2 were observed and studied as a function of temperature and excitation frequency. These signals gradually evolve into a resonance mode of the uncompensated ferromagnetic moments in the antiferromagnetically (AFM) ordered state below TN≈11K and 98 K for La2ZnIrO6 and La2CoIrO6, respectively. The temperature evolution of the La139 NMR spectra and the nuclear spin relaxation T1-1(T) reveal three distinct regimes for La2BIrO6 (B=Co, Zn): (i) a paramagnetic state at high temperatures; (ii) a short-range AFM order in an extended temperature region above TN where the ESR modes first appear; (iii) magnetic long-range order at lower T. The extended short-range-ordered regime at temperatures much higher than TN constitutes evidence of substantial magnetic frustration which appears to be a salient feature of both studied materials.