A slang term used to describe a quarterback that drops back to throw the football, but takes too much time to deliver the ball to a receiver (and taps the ball too much).
The phrase comes from the popular position of holding a newborn like a football and actually burping a baby. In football it refers to that constant tapping of the football with the quarterback's non-throwing hand.
Quarterbacks that 'burp the baby' too much means they usually hold on to the ball too long as opposed to just dropping back and throwing the ball. Generally speaking, a quarterback has 2-3 seconds to throw the ball before defensive pressure gets to him.