There is increasing theoretical and experimental evidence indicating that small-scale domain structure and dynamical heterogeneity develop in lipid membranes as a consequence of the the underlying phase transitions and the associated density and composition fluctuations. The relevant coherence lengths are in the nano-meter range. The nano-scale structure is believed to be important for controlling the activity of enzymes, specifically phospholipases, which act at bilayer membranes. We propose here a lattice-gas statistical mechanical model with appropriate dynamics to account for the non-equilibrium action of the enzyme phospholipase A₂ which hydrolyses lipid-bilayer substrates. The resulting product molecules are assumed to induce local variations in the membrane interfacial pressure. Monte Carlo simulations of the non-equilibrium properties of the model for one-component as well as binary lipid mixtures show that the enzyme activity is modulated by nano-scale lipid-domain formation in the lipid bilayer and lead to a characteristic lag-burst behavior. The simulations are found to be in semi-quantitative agreement with experimental data.