Quantum simulation on emerging quantum hardware is a topic of intense interest. While many studies focus on computing ground-state properties or simulating unitary dynamics of closed systems, open quantum systems are an interesting target of study owing to their ubiquity and rich physical behavior. However, their nonunitary dynamics are also not natural to simulate on digital quantum devices. Here, we report algorithms for the digital quantum simulation of the dynamics of open quantum systems governed by a Lindblad equation using adaptations of the quantum imaginary-time evolution algorithm. We demonstrate the algorithms on IBM Quantum's hardware with simulations of the spontaneous emission of a two-level system and the dissipative transverse field Ising model. Our work advances efforts to simulate the dynamics of open quantum systems on quantum hardware.