5.56 is designed to a higher maximum pressure and overall cartridge length than .223. Exterior case dimensions are identical. Intended bullets are heavier, and therefore longer, which might result in the bullet being jammed into the rifling. This would cause even higher chamber pressures.
You might be best to ask Savage if their barrels will handle the extra pressure if you were to rechamber. But rechambering should be really easy, since you only really have to increase the throat length.