In this paper, the effects on drying time (Y1), the color difference (Y2), unit energy consumption (Y3), polysaccharide content (Y4), rehydration ratio (Y5), and allantoin content (Y6) of yam slices were investigated under different drying temperatures (50–70 °C), slice thicknesses (2–10 mm), and radiation distances (80–160 mm). The optimal drying conditions were determined by applying the BP neural network wolf algorithm (GWO) model based on response surface methodology (RMS). All the above indices were significantly affected by drying conditions (p < 0.05). The drying rate and effective water diffusion coefficient of yam slices accelerated with increasing temperature and decreasing slice thickness and radiation distance. The selection of lower temperature and slice thickness helped reduce the energy consumption and color difference. The polysaccharide content increased and then decreased with drying temperature, slice thickness, and radiation distance, and it was highest at 60 °C, 6 mm, and 120 mm. At 60 °C, lower slice thickness and radiation distance favored the retention of allantoin content. Under the given constraints (minimization of drying time, unit energy consumption, color difference, and maximization of rehydration ratio, polysaccharide content, and allantoin content), BP-GWO was found to have higher coefficients of determination (R2 = 0.9919 to 0.9983) and lower RMSEs (reduced by 61.34% to 80.03%) than RMS. Multi-objective optimization of BP-GWO was carried out to obtain the optimal drying conditions, as follows: temperature 63.57 °C, slice thickness 4.27 mm, radiation distance 91.39 mm, corresponding to the optimal indices, as follows: Y1 = 133.71 min, Y2 = 7.26, Y3 = 8.54 kJ·h·kg−1, Y4 = 20.73 mg/g, Y5 = 2.84 kg/kg, and Y6 = 3.69 μg/g. In the experimental verification of the prediction results, the relative error between the actual and predicted values was less than 5%, proving the model’s reliability for other materials in the drying technology process research to provide a reference.