[AngelicaDeAlquezar]

I think the most important part of this expression depends on "en saco roto" more than in the fact of lying.

"Echar/caer en saco roto" means that something doesn't have an expected effect.

Es inútil aconsejar a Juan. Lo que le digas, cae en saco roto.
It's useless to give a piece of advice to Juan. He never listens to anything you say to him.

No eches en saco roto las advertencias del doctor, podrías regresar al hospital.
Listen to the warnings of the doctor; you could go back to hospital.