Cu2IrO3 is among the newest layered honeycomb iridates and a promising candidate to harbor a Kitaev quantum spin liquid state. Here, we investigate the pressure and temperature dependence of its structure through a combination of powder x-ray diffraction and x-ray absorption fine structure measurements, as well as ab initio evolutionary structure search. At ambient pressure, we revise the previously proposed C2/c solution with a related but notably more stable P21/c structure. Pressures below 8 GPa drive the formation of Ir-Ir dimers at both ambient and low temperatures, similar to the case of Li2IrO3. At higher pressures, the structural evolution dramatically depends on temperature. A large discontinuous reduction of the Ir honeycomb interplanar distance is observed around 15 GPa at room temperature, likely driven by a collapse of the O-Cu-O dumbbell bonds. At 15 K, pressures beyond 20 GPa first lead to an intermediate phase featuring a continuous reduction of the interplanar distance, which then collapses at 30 GPa across yet another phase transition. However, the resulting structure around 40 GPa is not the same at room and low temperatures. Remarkably, the reduction in interplanar distance leads to an apparent healing of the stacking faults at room temperature, but not at 15 K. Possible implications on the evolution of electronic structure of Cu2IrO3 with pressure are discussed.