Cac, Your metabolism is elevated while you are running and doing cardio exercises and stays elevated for a period after you exercise, but it doesn't stay elevated. Building muscle will boost your metabolism though.
You can't out-exercise poor eating habits.
HIIT (high intensity interval training) first thing in the morning has been shown to increase your metabolic rate throughout the day; however, the slow, steady cardio that is performed at the "fat burning" heart rate is not said to continue after the exercise has ceased.
I second what Nelie said about building muscle! That's a sure thing.