Phase transitions and thermodynamic properties of the frustrated J1-J2-J3 Ising model on the honeycomb lattice are investigated by using the cluster mean-field theory and Monte Carlo simulations. We investigate the role of thermal fluctuations in the model with ferromagnetic exchange interactions between first and third neighbors and antiferromagnetic second-neighbor couplings. Our findings indicate that these competing interactions can lead to three forms of long-range order and different scenarios concerning phase transitions. For weak third-neighbor interactions, strong frustration effects bring the ordering temperature to absolute zero and lead to residual entropy. Moreover, thermodynamic signatures of frustration such as a round maximum in the specific heat can be found above the ordering temperature. The model also exhibits continuous and discontinuous order–disorder phase transitions, hosting tricriticality for weak and moderate strengths of third-neighbor couplings. At strong enough third-neighbor couplings, discontinuous order–disorder phase transitions are suppressed and a bicritical point can be observed. Our findings are discussed in connection with recent experimental results for Ising spin systems on the frustrated honeycomb lattice, including van der Waals magnets.