Does sea salt make the pus come out of lip piercings?

Not exactly. If its really pus, you need to see a doctor because pus is an indication of infection. If you're not experiencing any swelling, itchiness, or redness in the area, it's just lymph. Once the piercing goes through it's initial healing phase, the lymph production will cut back a lot. Sea salt helps the piercing heal, so you should definitely use it.