We have synthesized single crystals of Na2(Ir1-xTix)O3 and polycrystals of Li2(Ir1-xTix)O3 and studied the effect of magnetic depletion on the magnetic properties by measurements of the magnetic susceptibility, specific heat, and magnetocaloric effect at temperatures down to 0.1 K. In both systems, the nonmagnetic substitution rapidly changes the magnetically ordered ground state into a spin glass, indicating strong frustration. While for the Li system the Weiss temperature ΘW remains unchanged up to x=0.55, a strong decrease |ΘW| is found for the Na system. This suggests that only for the former system magnetic exchange beyond nearest neighbors is dominating. This is also corroborated by the observation of a smeared quantum phase transition in Li2(Ir1-xTix)O3 near x=0.5, i.e., much beyond the site percolation threshold of the honeycomb lattice. © 2014 American Physical Society.