Since "release" is the term that the community uses for lethal injection, we know that the people who are released die. What happens to their physical bodies after death is not discussed in the book, and whether there is an after-life certainly is not. So, physically and spiritually the reader does not have any indication of where these people go after release.
The community in The Giver does not discuss death itself. They never use the word, so these conversations about what happens next do not occur either.
Release is seen differently depending on who experiences it. For example, when an elderly person is released, it is a happy event, a celebration. But, the people at the celebration do not see the actual release.
A ceremony is held, which includes a "telling of the life," a toast, an anthem, a good-bye speech from the individual to be released (where appropriate), some farewell speeches from those who know him or her, then a walk through the door to the Releasing Room" (http://www.enotes.com/topics/giver/themes)
But release can also be a punishment for those who break rules of the community AND newborn children (or those under the age of 1) may be released for a variety of reasons. Gabe is scheduled to be released because he has not progressed as the rate of a normal child. He does not grow fast enough and cannot sleep well unless he is with Jonas. Towards the end of the book, Jonas's father also releases a newborn twin because the community does not allow twins, so the nurturers must weigh and measure the twins when they are born and release the smaller one.
In the case of the elderly, we do not know where their bodies go after release, as there is never discussion of burial or graveyards or cremation.
For the newborn twin, his body is put into a box and sent down what appears to be a trash chute. No explanation is given as to where the chute empties or what happens to the baby's body.