Is it healthy to go run while my chest and arm muscles are still sore from working out?

Yes in my opinion: I think it is a great idea to alternate muscle groups in the course of a fitness program. The soreness is likely attributable to anaerobic metabolism or Lactic Acid buildup. If it does not resolve within a few days, it could be due to injury. In the meantime i see no contraindications to aerobic or lower body workouts performed in a reasonable fashion.