If Oedipus' story gets another twist in the story of his lineage and it turns out his actual parents are yet another set of people that he goes to visit at the end of the play, we have the makings of a macabre comedy.
All the previous answers are intriguing and suggestive. I'm sure this play could be turned into a comedy with some effort, and with a broad (and dark) enough definition of comedy. Interestingly, there are already some moments in the play that already seem potentially comic, especially if the lines were delivered in the right tone of voice. Consider, for example, this exchange between Tiresias and Oedipus:
OEDIPUS: You dare to utter shameful words like this?
Do you think you can get away with it?
TEIRESIAS: I am getting away with it.
All it would take to begin moving this exchange in a comic direction would be to have Tiresias give his line the right inflection: "I am getting away with it!!!" (Delivered as if to say to Oedipus, "Isn't it obvious? Are you stupid?")
Or consider this exchange, which quickly follows the one just quoted:
OEDIPUS: Who taught you this?
It could not have been your craft.
TEIRESIAS: You did.
I did not want to speak, but you incited me.
All Tiresias would need to do to begin moving his reply in the direction of comedy would be to give the right inflection to "you" each time he says it. The exchanges between Tiresias and Oedipus offer a number of such opportunities, as do the exchanges between Oedipus and Cleon. It would be interesting, in fact, to see if the whole play could be given a comic spin even as presently written. (I doubt that this would be possible, but I haven't tried.) The comedy would have to be absurdist, Monty-Pythonish (think Life of Brian), but that's a popular kind of comedy these days.
A serious rewrite would be needed but if the angst could be taken out of the death and tragedy; if the death in the play could be reduced to the level of a Charlie Chan murder mystery; if his punishment of blindness could be scripted as the metaphorical sort; if Jocasta's pregnancy and suicide could be likewise reduced to some metaphorical representation; and if his exile might be of a sort that is more in the line of taking a job at a resort in the Bahamas or on a cruise ship, then maybe--only maybe--the end result might produce a comedy. Only maybe.
Parody or farce is the way to go with any comedic presentation of Oedipus Rex since these forms of comedy exaggerate the characteristics of the personages in a drama. With all the discussion over the years about the Oedipal Complex, the marriage of Oedipus to his mother can certainly be portrayed in a farcical manner. Many other dramatic actions can easily be exaggerated to the level of farce; for example, Oedipus's anger at his real father, and his throwing of him from his horse; and Oedipus's refusal to believe the truth from Teiresias and Creon. The great playwright of the sixteenth century in France, Moliere, certainly did an excellent job parodying the majestic kings such as the Sun King, Louis XIV, so there is no reason to think that the mighty Oedipus Rex cannot also be parodied.
I like this question. Whilst there is something obviously tremendously tragic abotu Oedipus and his gradual realisation that it is he who is the murderer that he seeks, a darkly ironic person could find cause for black humour in the dramatic irony, as he desperately looks for the murderer only to be blind to the fact that it is himself he looks for. You could compare this to a pantomime where the actor tries to find another actor and the audience is left screaming "He's behind you." Maybe it is just my twisted and warped mind that could see a comic element to this tragedy though!
If you consider a definition of comedy that suggests "happy ending" rather than ha ha laughter, then I think you could be on to something. If Oedipus and Jocasta could find a way to make amends for their actions and find a way for the family to live in peace, then I think you could have a happy ending. The solving of the mystery/curse does save the country, so that is a step in the right direction. Perhaps they could appeal to the gods for guidance or go to the Oracle and make amends there.
I agree with Lori, that the basic storyline (and of course ending) simply isn't a laughable subject. However, I think about all of the "dark comedies" which are so prevalent in film today, and I actually don't think the idea of turning the story of Oedipus into a comedy would be entirely impossible. Again, the focus would have to be taken away from pitying Oedipus (and his family) and instead, using them as an example of something so stupid or ridiculous or fault-full that the story takes on a cultural or idiosyncratic focus. I think Auntlori has a great idea with the idea of putting the point of view more on the gods than on the humans.
I can't imagine wanting to do such a thing, first of all. A look at the skeleton of the story doesn't give the audience much to laugh about, it seems to me. Two dead parents, a son/husband who gouges his eyes out, and four children who will suffer the consequences of their father's actions--not to mention a country which is dying. While there are some good moments of thoughtful laughter as the irony unfolds, there just isn't much humor in this drama. There are two areas which might offer a little more room for humor, both through the use of exaggeration. One is showing Oedipus as an even more pompous and ridiculously arrogant King; the other is to show a gathering of the gods dicussing or rejoicing or laughing--something--over the pitiful attempts of Oedipus to escape his fate.