In this work, we utilize micropipette aspiration and fluorescence imaging to examine the material properties of lipid vesicles made from mixtures of palmitoyloleoylphosphocholine (POPC) and dipalmitoylphosphatidylcholine (DPPC). At elevated temperatures/low DPPC fractions, these lipids are in a miscible liquid crystalline (L(alpha)) state, whereas at lower temperatures/higher DPPC fractions they phase-separate into L(alpha) and gel phases. We show that the elastic modulus, K, and critical tension, tau(c), of L(alpha) vesicles are independent of DPPC fraction. However, as the sample temperature is increased from 15 degrees C to 45 degrees C, we measure decreases in both K and tau(c) of 20% and 50%, respectively. The elasticity change is likely driven by a change in interfacial tension. We describe the reduction in critical tension using a simple model of thermally activated membrane pores. Vesicles with two-phase coexistence exhibit material properties that differ from L(alpha) vesicles including critical tensions that are 20-40% lower. Fluorescence imaging of phase coexistent POPC/DPPC vesicles shows that the DPPC-rich domains exist in an extended network structure that exhibits characteristics of a solid. This gel network explains many of the unusual material properties of two-phase membranes.