Distributed model predictive control (DMPC) concerns how to online control multiple robotic systems with constraints effectively. However, the nonlinearity, nonconvexity, and strong interconnections of dynamic system models and constraints can make the real-time and real-world DMPC implementations nontrivial. Reinforcement learning (RL) algorithms are promising for control policy design. However, how to ensure safety in terms of state constraints in RL remains a significant issue. This paper proposes a barrier function-based safe reinforcement learning algorithm for DMPC of nonlinear multi-robot systems under state constraints. The proposed approach is composed of several local learning-based MPC regulators. Each regulator, associated with a local system, learns and deploys the local control policy using a safe reinforcement learning algorithm in a distributed manner, i.e., with state information only among the neighbor agents. As a prominent feature of the proposed algorithm, we present a novel barrier-based policy structure to ensure safety, which has a clear mechanistic interpretation. Both simulated and real-world experiments on the formation control of mobile robots with collision avoidance show the effectiveness of the proposed safe reinforcement learning algorithm for DMPC.