We study a one-dimensional quasiperiodic tight-binding model with simultaneous off-diagonal (hopping) and diagonal (onsite) modulations. Using the inverse participation ratio and the wave-packet centroid, we construct localization–delocalization phase diagrams for both equilibrium and nonequilibrium steady states. We analyze the robustness of initial-state properties under dissipation and characterize dissipation-induced localization–delocalization transitions (and their reversals) in detail. Trace-distance dynamics provide evidence for a quantum Mpemba effect: states prepared farther from the steady state can relax faster than states initialized closer to it. We propose a starting line hypothesis to explain the presence or absence of this effect in different parameter ranges of our selected system. Through the study of wave packet transport under the same dissipative system, we analyzed the applicable conditions of the starting line hypothesis and summarized a way to construct the Mpemba effect based on this. These results advance the understanding of steady-state phase transitions and relaxation dynamics in dissipatively driven quasiperiodic systems.