Quantum simulation of many-body systems offers a powerful approach to exploring collective quantum dynamics beyond classical computational reach. Although spin and fermionic models have been extensively simulated on digital quantum computers, the simulation of bosonic systems on programmable quantum processors is often hindered by the intrinsically large Hilbert space of bosonic modes. In this work, we study the digital quantum simulation of bosonic modes using the Holstein-Primakoff (HP) transformation and implement this protocol on a cloud-based superconducting quantum processor. Representative bosonic dynamics, including the driven harmonic oscillator, bosonic squeezing, and the Jaynes-Cummings (JC) model, are simulated on quantum hardware. Using data obtained from the quantum simulations, we systematically examine the interplay between algorithmic and hardware-induced errors to identify optimal simulation parameters. The dominant algorithmic errors arise from the finite number of qubits used in the HP mapping and the finite number of Trotter steps in the time evolution, while hardware errors mainly originate from entangling-gate infidelity, decoherence, and readout errors. We demonstrate two-excitation JC dynamics and squeezed-state generation involving qubit-qubit interactions under the HP transformation, extending the simulations beyond the simple single-excitation regime. Furthermore, we show that quantum error mitigation substantially improves the fidelity of the quantum simulations on current hardware. Our results demonstrate the feasibility and limitations of HP-based digital simulation of bosonic degrees of freedom on current quantum processors and provide a framework that can be extended to more complex spin-boson and multimode cavity models.