The neurohypophysis, also known as the posterior pituitary, is the rearmost portion of the pituitary gland. Its functions are closely integrated with those of the nervous system. The posterior pituitary is responsible for the production of ADH, or antidiuretic hormone, and of oxytocin, so a person who lost this tissue would be unable to produce either hormone.
ADH is vital in maintaining the fluid balance in the body, as it slows the filtration rate of the kidneys and conserves water. A person with an insufficient amount of ADH will develop Central Diabetes Insipidus.
Oxytocin's role in males is not well understood, but it is believed that the hormone assists in the transport of sperm within the male reproductive tract, and the hormone is also present in semen, so a male unable to produce this hormone might be infertile.
Oxytocin's functions in women is much better understood; in the female, it stimulates uterine contractions during birth, stimulates milk letdown for nursing, and is believed to assist in the development of maternal bonding and behaviors. A woman who could not produce this hormone would probably need to deliver a child by Cesarean, and would be unable to nurse it.