Nickel (Ni) and cobalt (Co) are strategic metals that have found applications in a wide range of metallurgical and industrial uses. In this study, the dissolution of a mixed nickel–cobalt hydroxide precipitate using organic acids (citric, oxalic, and malic acid) was investigated. Citric acid was found to be the best leaching agent yielding the following dissolution rates: 91.2% Ni, 86.8% Co, and 90.8% Mn. Oxalic acid resulted in low dissolution, which is likely due to the formation of insoluble metal oxalates. The impact of acid concentration, leaching time, and temperature on metal dissolution was systematically examined. The optimal dissolution conditions were identified as 0.5 M citric acid at 30 °C for 30 min, utilizing a 1/20 solid/liquid ratio and a stirring speed of 400 revolutions per minute (rpm). The attempt to use oxidants, such as potassium permanganate (KMnO₄) and hydrogen peroxide (H₂O₂), to achieve selective dissolution in an organic acid environment was not successful, which was different from that in the sulfuric acid case. As for the leaching kinetics in the organic acids, it seems that the leaching of Ni correlates with the Shrinking Core Model, specifically regarding porous-layer diffusion control. Based on the experimental results, the activation energy for the leaching of Ni was estimated to be 3.1 kJ/mol.