There is also significance in the final scene of Oedipus the King that extends beyond the boundaries of this one play. It was part of a trilogy, and like any good serial, needed to set up the plays to follow. In this case they were: Oedipus at Colonus and Antigone.
Both plays are foreshadowed here in the final scene. First, Oedipus begs to be allowed to leave Thebes, and Creon gives him permission, hinting that the Gods also agree. That sets up Oedipus at Colonus.
Oedipus also ends the play by asking to hold his daughters, Antigone and Ismene, daughters of himself and Jocasta (his mother/wife). It is significant that Oedipus is more worried about these girls and what will befall them than his sons, since the play Antigone is very focused on the dilemmas that especially Antigone faces. Creon commands him to leave the children, and though he does not want to, Oedipus complies. Creon reiterates Oedipus' fall from power with the lines (1523-24):
Do not seek to be master in everything,
for the things you mastered did not follow you throughout your life.
So the future lives of the children of Oedipus, cursed by their parents' incest, are hinted at here, as is the power (that Creon has claimed throughout the entire play not to want) that now falls on Creon as the ruler in Thebes. Creon and his role as ruler plays a very important part in the play Antigone.
The closing scene in Oedipus Rex is tragic and sets the stage for the punishment of Oedipus. The son/husband of the now-dead Jocasta has discovered the truth of his horrific sins and leaves his home for the last time as a blind and bloodied man. He has blinded himself literally as punishment for his figurative blindness regarding his role in his country's curse. He asks the new king, his uncle/brother-in-law Creon, for several things. First, he asks to see his children again. After tracing their faces with his hands, he asks Creon to take care of his girls; he understands they will have extra difficulties in life once his abomination has been made known. Second, he asks Creon to exile him, which Creon promptly does.
The significance of this scene is twofold, for me. One, it fulfills all the curses which Oedipus so arrogantly places on whoever killed Lauis, thereby bringing the curse of the gods on his own city. This is the ultimate irony of the play. Second, it is a vivid picture of what happens when one defies the gods and attempts to control his own fate.
Oedipus Rex is what many consider to be the perfect tragic play; Oedipus, the title character, perfectly embodies Aristotle's description of a tragic hero, and the action of the play makes it what Aristotle defined as a true tragedy.
In the closing scene of Oedipus Rex, Oedipus, who has recently learned that he has unknowingly killed his father and married his mother, blinds himself as a form of self-punishment and prepares to live the rest of his life in exile.
The Chorus closes the play with these final lines:
People of our country Thebes, behold this Oedipus,
who knew the famous riddle and was a most powerful man,
whose fortunes all the citizens watched with emulation,(1555)
how deep the sea of dire misfortune that has taken him!
Therefore, it is necessary to call no man blessed
as we await the final day, until he has reached
the limit of life and suffered nothing grievous.
This final scene is significant because it portrays Oedipus as a tragic hero and shows audiences that not even a noble man such as Oedipus is free of sin or wrongdoing. After his reversal of fortune, Oedipus accepts responsibility for his actions (killing his father and marrying and having children with his mother) by blinding himself so that he won't have to "see" his sin--his children--any more. Further, he prepares to live a life of exile since it is the punishment he vowed to impose upon the person who killed Laius (before he understood that he himself was the killer).
Essentially, the Chorus, in the last lines of the play, is observing that no humans who are living are "blessed," as even the most powerful and respected people can fall victim to fate at any time.