Membrane-associated proteins are likely to contribute to the regulation of the phase behavior of mixed lipid membranes. To gain insight into the underlying mechanism, we study a thermodynamic model for the stability of a protein-decorated binary lipid layer. Here, proteins interact preferentially with one lipid species and thus locally sequester that species. We aim to specify conditions that lead to an additional macroscopic phase separation of the protein-decorated lipid membrane. Our model is based on a standard mean-field lattice-gas description for both the lipid mixture and the adsorbed protein layer. Besides accounting for the lipid-protein binding strength, we also include attractive lipid-lipid and protein-protein interactions. Our analysis characterizes the decrease in the membrane's critical interaction parameter as a function of the lipid-protein binding strength. For small and large binding strengths we provide analytical expressions; numerical results cover the intermediate range. Our results reiterate the crucial importance of the line tension associated with protein-induced compositional gradients and the presence of attractive lipid-lipid interactions within the membrane. Direct protein-protein attraction effectively increases the line tension and thus tends to further destabilize the membrane.