There are some specific actions that key characters take regarding death that have a huge effect on the fulfillment of the Oracle's prediction that Oedipus would "kill his father and marry his mother."
First Jocasta confesses that she has beaten the Oracle at it's own game by taking a child that she had and ordering a shepherd to abandon it and leave it to die. In this way, she would avoid the prediction by taking matters (and Fate) into her own hands -- if it had worked.
When Oedipus heard the prophecy that he would kill his father and marry his mother, he left his home with his "father" Lepidus, and travelled to escape this prediction. He met a man on the road as stubborn as himself, and when neither would move aside to let the other pass on the narrow road, he, in a rage, killed this man and went on his way.
Of course, it is discovered later that Jocasta's baby was saved by the same shepherd she gave it to, when he gave the child to a shepherd from Lepidus' land, who, in turn, gave the child to Lepidus' wife, who longed for a baby. It is also discovered that the man that Oedipus killed was Laius, Jocasta's husband and Oedipus' father.
When the facts of these deaths and escaped deaths are revealed, it is clear that the oracle, despite the maneuvers of the humans in the play, has indeed come true. So, death can be seen in the play as a fact of Fate, something ordained by the gods that, try as one human might, cannot be outwitted. This could definitely be seen as one part of the sobering and tragic vision of the play.
In the end, the presence of death is more of a reminder that Oedipus' hubris is really unfounded. Oedipus believes that his own sense of self and pride can overcome everything, including the plague that has fallen on his own people. His belief in his own sense of self, wanting to figure out the mystery of the plague, of death, is something that drives him in his quest. It is this sense of hubris and faith in himself that compels him to try to challenge the forces of death and push the limits of mortality in general that brings about his tragic condition. The ideas of seeking to defy the roles and functions of fate and what is destiny as well as the power of death that grips both his people and eventually him are realities that Oedipus does not fully accept or even comprehend. It is no wonder why that the ending of the drama is one where he understands these forces with stunning brutality as "sight" is restored to him about these realities in his own finite life.