Oedipus shows weakness in several areas. His hubris (conceited pride) is discussed in the answer above and can be seen as his tragic flaw and most significant weakness. However, Oedipus shows weakness elsewhere as well.
In dealing with Creon, Oedipus acts unreasonably and is unable to control his anger. Paranoia, anger and impatience all propel Oedipus into a rash conclusion.
Oedipus feels threatened by Creon and believes that he covets the throne...
When others, like Tiresias, Jocasta and Creon attempt to advise restraint, Oedipus grows impatient and angry with them. This lack of self-control is a weakness.
Oedipus is desparate to know the truth about his situation. His emotion overwhelms his reason, leading to self-control issues, paranoia and anger.
Oedupus' greatest weakness is over confidence; a kind of excessive pride (hubris) that leads to his downfall. For instance, when he first heard the prophecy that he would, one day, murder his father and marry his mother, he tried to circumvent this prophecy by running away from his home town, unaware that the people he believed were his parents, were really his adoptive parents. Ironically, in seeking to circumvent this prophecy, he succeeds in fulfilling it. For, on his journey out of town, he encounters his real father, has an argument with him and ends up killing him. Later he arrives at his place of birth and marries the widow of the man he had killed, not knowing her connection to him or to himself. Though it seems entirely reasonable that a person should seek to avoid fulfilling such a terrible prophecy, Sophocles is trying to make the point that one cannot seek to outwit the gods. In seeking to do so, Oedipus was displaying excessive pride in his own abilities.
Later on, when Thebes was beset by famine, he tries to get at the root of the problem, and find out the reason why the gods were angry with Thebes. Despite being warned by Creon, by Tirisias the prophet, by the shepherd who had rescued him as a baby and finally by his wife, who had begun to get an inkling of the truth, not to pursue the matter any further, he does do so. While this does show strength of character - that he is willing to face the truth, however unpalatable - it also reveals that he is over confident, stubborn and unwilling to listen to advice from others. His strength thus becomes his weakness and leads to the destruction of everything he holds dear, and to his ultimate destruction