Relaxation phenomena in two-dimensional (2D) plasma crystals have been investigated, including both the self-consistent electric charge of dust particles and the electron and ion velocity distribution functions, by means of a modified 2D particle-in-cell (PIC) method. The results obtained show that the mutual interaction of dust particles in such crystals leads to special properties of the background electrons and ions due to their selective collection by dust particles during the relaxation. These electrons and ions can behave as non-ideal components of dusty plasmas in plasma crystals even in cases where their numbers in the Debye cube are large. This effect is caused by their intensive charge exchange with dust particles, which provides dusty plasmas with the status of open statistical systems. The selective collection of electrons and ions by dust particles also causes their deviation from the initial equilibrium as well as the non-equilibrium evolution of the self-consistent electric charge of the dust particles. Relaxation phenomena in plasma crystals have to be taken into account in all cases of strong changes of plasma parameters, for example due to strong oscillations and waves in these crystals.