Cellular membranes are a heterogeneous mix of lipids, proteins and small molecules. Special groupings of saturated lipids and cholesterol form a liquid-ordered phase, known as `lipid rafts,' serving as platforms for signaling, trafficking and material transport throughout the secretory pathway. Questions remain as to how the cell maintains heterogeneity of a fluid membrane with multiple phases, through time, on a length-scale consistent with the fact that no large-scale phase separation is observed. We have utilized a combination of mechanical modeling and in vitro experiments to show that membrane morphology can be a key player in maintaining this heterogeneity and organizing such domains in the membrane. We demonstrate that lipid domains can adopt a flat or dimpled morphology, where the latter facilitates a repulsive interaction that slows coalescence and tends to organize domains. These forces, that depend on domain morphology, play an important role in regulating lipid domain size and in the lateral organization of lipids in the membrane.