The AC system itself operates at much lower temperatures, with refrigerant typically around 30-50°F (-1° to 10°C). However, this is a different system and not related to the engine's operating temperature.
If the engine is overheating (above 220°F/104°C), that's a serious problem regardless of AC use and requires immediate attention. The AC compressor might put a slightly higher load on the engine, but it shouldn't cause overheating unless there's already an issue with the cooling system.