Food packaging plays an essential role in preserving food quality. However, heavy metals found in packaging materials—whether intentionally incorporated or not—can migrate into food. This study aims to evaluate the migration of specific heavy metals (Ba, Co, Cu, Zn, Al, Ni, Li, Fe, Pb, Cd, Cr, Sb) from plastic food packages (films and bags) obtained from various materials (PE, PP, PVC, composite materials) into food simulant B (3% acetic acid) using inductively coupled plasma mass spectrometry (ICP-MS). Migration tests was conducted according to EU regulations, using OM2 conditions (10 days at 40 °C). The obtained results were lower than the specific migration limits set by EU Regulation no. 10/2011 (Annex II). Both carcinogenic and non-carcinogenic risk assessments were carried out based on the specific migration data, estimating the exposure, average daily dose (ADD), hazard quotient (HQ), hazard index (HI), cancer risk (CR), and total cancer risk (TCR). The exposure values were found to be below the recommended tolerable daily intake (TDI) levels for each metal tested. Both HQ and HI values were under the limit value of 1. The average total cancer risk was 1.73 × 10−4, indicating that approximately 1.73 consumers out of 10,000 may develop a type of cancer due to chronic exposure to the tested metals. These results highlight the importance of continuous monitoring of chemical migrants from food contact materials.