Dry birth" is really a misnomer because your body continues to make amniotic fluid. This term probably originated in the days of granny and traditional midwifery. Fluid is constantly produced -- about a cup an hour. Even if there is no fluid release with the birth, there is usually fluid behind the baby.
The amniotic fluid probably serves no real lubricative function, but absence or low levels of fluid is a cause for concern for other reasons.
Oligohydramnios, or too little fluid, is common in the postdate pregnancy. It can reflect the fetus' diminished production of urine and the deterioration of the fetal-maternal circulation.
In postdate pregnancy, the risk of fetal jeopardy is higher from placental insufficiency and also from the higher risk of passage of meconium. When there is not much fluid, the meconium becomes very particulate and thick. Aspiration into the baby's lungs before and after birth is a major risk.
Because fluid volume does diminish after term, instillation of warm saline can be done if this lack of fluid seems to be causing distress in labor or if the baby has passed meconium and the midwife or doctor feels that it would be helpful to try to dilute it. This procedure is called "amnioinfusion" and it can be very beneficial.