Seaweed and fish have slippery outer surfaces because of the secretion of a layer of mucus. Navier slip arises when the component of the tangential velocity at a wall is proportional to the strain. The hydrodynamics of a three-dimensional flexible plate with Navier slip was explored by using the immersed boundary method in an effort to scrutinize the effects on plate hydrodynamics of a slip boundary mimicking the mucus layers of seaweed and fish. For comparison, simulations with the no-slip condition were also performed. Two cases were chosen for simulation: a flexible plate with a fixed leading edge and a flexible plate with a heaving leading edge in a uniform flow. For the fixed plate, the velocity gradient and the total drag were determined to examine the influence of the slip surface. Drag was significantly reduced by the slip. The slip surface lessens the velocity gradient near the wall and suppresses the flapping motion. The drag reduction process was characterized by using the distributions of vorticity and pressure. The hydrodynamics of the heaving flexible plate with Navier slip was explored in terms of thrust generation. The flapping motion was mainly governed by the input heaving condition and a large form drag was exerted on the flexible plate. The net thrust, input power, and Froude efficiency were determined as a function of the bending rigidity. A large net thrust for the heaving plate was generated by the slip. The velocity ratio was employed to interpret the correlation between the slip velocity and the flapping motion.