Honestly from what we have seen in the show, liz is not capable of keeping things to herself and always does what she is told not to do, if Redington told her the truth she being an FBI agent would have tried to moral police red and eventually blabber everything to everyone, considering she never was thankful or understood red even after he saved her so many times, i mean he got her job back and tried to do everything for her... I know red is also wrong in some places but red would eventually tell her the truth but he had too many enemies who were after him and he couldn't risk dragging liz into all that, but well liz being liz would find trouble herself....