Alfvén waves as excited in black hole accretion disks and neutron star magnetospheres are the building blocks of turbulence in relativistic, magnetized plasmas. A large reservoir of magnetic energy is available in these systems, such that the plasma can be heated significantly even in the weak turbulence regime. We perform high-resolution three-dimensional simulations of counter-propagating Alfvén waves, showing that an $E_{B_{⊥ }}(k_{⊥ }) ∝ k_{⊥ }^{-2}$ energy spectrum develops as a result of the weak turbulence cascade in relativistic magnetohydrodynamics and its infinitely magnetized (force-free) limit. The plasma turbulence ubiquitously generates current sheets, which act as locations where magnetic energy dissipates. We show that current sheets form as a natural result of nonlinear interactions between counter-propagating Alfvén waves. These current sheets form owing to the compression of elongated eddies, driven by the shear induced by growing higher-order modes, and undergo a thinning process until they break-up into small-scale turbulent structures. We explore the formation of current sheets both in overlapping waves and in localized wave packet collisions. The relativistic interaction of localized Alfvén waves induces both Alfvén waves and fast waves, and efficiently mediates the conversion and dissipation of electromagnetic energy in astrophysical systems. Plasma energization through reconnection in current sheets emerging during the interaction of Alfvén waves can potentially explain X-ray emission in black hole accretion coronae and neutron star magnetospheres.