There is no general birth rite or obligations across the Buddhist world. Practices vary from one country and culture to another.
In Sri Lanka and other Theravada countries, when a baby is born, monks might be invited to the house where they prepare a horoscope for the baby based on the time of birth. From the details of the horoscope, they decide the first name of the baby.
Within a month of the birth, the parents take the baby to the local temple and put him or her in front of the statue of the Buddha. They ask for the blessings of the Three Refuges - the Buddha, the dharma and the Sangha.
The parents make an offering to the temple in thanksgiving for the child.
Some Buddhists, for example in Thailand, tie sacred threads around the baby's wrists to greet the spirit 'Khwan' who looks after babies. When the baby is a month old, a monk may shave the baby's hair.
In Tibet when a baby is born the parents put flags on the roof of their house, and after a few days friends and relatives gather at the house bringing gifts of food and clothes. A monk will visit and a naming ceremony will take place after about a ten days. The baby does not leave the home for the first month, when it may be taken to the local monastery.
In Japan, a few days after birth the baby's room is purified. As in Tibet there is a naming ceremony at home about one week after the birth, and on the child's first visit to the temple the parents are presented with a scroll for recording the child's life events.
When the parents present the child in the temple, they are introducing the new child to the Buddha, and showing that they want their child to grow up understanding the Buddhist tradition and the Four Noble Truths - and to be able to put the Noble Eightfold Path into practice.