Did Oedipus need to be taught modesty through suffering? Are there any quotations that support this?

One account of tragedy argues that it follows a pattern in which the hero, who is otherwise a great and noble character, suffers from a tragic flaw of arrogance or overwhelming pride leading to his inevitable downfall. This account is somewhat of an oversimplification, as the hero is sometimes caught in a forced choice scenario—often due to an inherited curse—and whatever path the hero chooses will lead to a downfall.
The important thing to note is that human pride or arrogance in Greek tragedy is often seen as an affront to the gods. Because the Greek pantheon is anthropomorphic, the difference between god and human is one of degree (of skill, power, strength, etc.), and thus human overreach is seen as a mortal challenging or impinging on the domain of the gods.

The tyrant is a child of Pride...
Until from his high crest headlong
He plummets to the dust of hope.

A tyrant was a ruler by popular acclaim or by force, while a king was an hereditary ruler. Because the people and Oedipus himself do not know Oedipus's birth circumstances until the end of the play, he is a tyrant; only when his parentage is discovered do we realize he was actually a king as well as tyrant. This quotation suggests that tyrants are proud and powerful and, because of that, endure a spectacular downfall.

Count no man happy till he dies, free of pain at last.

Even when one is wealthy or powerful, one is not immune to the power of the gods and fortune and should not become complacent.

If a man's contemptuous, and goes along with acts and speaks without respect for what is right and doesn't revere statues of gods, then let a sorry fate destroy him.

This quotation suggests that if someone is arrogant and does not display adequate respect for the gods, the gods will teach him modesty by destroying him.