Reported in the paper are results of unsteady three-dimensional direct numerical simulations of laminar and turbulent, lean hydrogen-air, complex-chemistry flames propagating in forced turbulence in a box. To explore the eventual influence of thermodiffusive instability of laminar flames on turbulent burning velocity, (i) a critical length scale Λ_{n} that bounds regimes of unstable and stable laminar combustion is numerically determined by gradually decreasing the width Λ of computational domain until a stable laminar flame is obtained, and (ii) simulations of turbulent flames are performed by varying the width from Λ<Λ_{n} (in this case, the instability is suppressed) to Λ>Λ_{n} (in this case, the instability may grow). Moreover, simulations are performed either using mixture-averaged transport properties (low Lewis number flames) or setting diffusivities of all species equal to heat diffusivity of the mixture (equidiffusive flames), with all other things being equal. Obtained results show a significant increase in turbulent burning velocity U_{T} when the boundary Λ=Λ_{n} is crossed in weak turbulence, but almost equal values of U_{T} are computed at Λ<Λ_{n} and Λ>Λ_{n} in moderately turbulent flames characterized by a Karlovitz number equal to 3.4 or larger. These results imply that thermodiffusive instability of laminar premixed flames substantially affects burning velocity in weak turbulence only, in line with a simple criterion proposed by Chomiak and Lipatnikov (Phys. Rev. E 107, 015102, (2023)10.1103/PhysRevE.107.015102).