A new inequality for a nonlinear surface layer integral is proved for minimizers of causal variational principles. This inequality is applied to obtain a new proof of the positive mass theorem with volume constraint. Next, a positive mass theorem without volume constraint is stated and proved by introducing and using the concept of asymptotic alignment. Moreover, a positive quasilocal mass and a synthetic definition of scalar curvature are introduced in the setting of causal variational principles. Our notions and results are illustrated by the explicit examples of causal fermion systems constructed in ultrastatic spacetimes and the Schwarzschild spacetime. In these examples, the correspondence to the ADM mass and similarities to the Brown–York mass are worked out.