The lips are an important part of the human body, so it is only logical, that they also form part of every day colloquial English.

to pay lip service =  only to agree with words, not with actions

e.g. Don’t just pay lip service, do something about it.

to keep up stiff upper lip = showing persitence in the face of adversity

e.g. Don´t let this get you down, keep a stiff upper lip.

to give sombody a lip = being disrespectful verbally

e.g. Don’t give me no lip, show some frigging respect.

loose lips, sink ships = careless talk can cause damage

e.g. Just shut up, don’t you know that loose lips sink ships.