How do the ancient heroes Oedipus (Sophocles), Lanval (Marie de France), and Sir Gawain (Sir Gawain and the Green Knight) fit into our modern conception of heroism?

The idea of the "hero" is a little difficult to pin down. Most people tend to think of a hero as someone of great physical courage who is not afraid to risk death in order to save others or to protect a noble cause. Your question refers to a "modern" conception of heroism, which might have more to do with moral courage or with the hero's actions stemming from some sort of internal conflict, perhaps between the "good" and "bad" sides of their nature.
The three texts you mention all present "heroes" that deviate somewhat from the conventional idea of heroism.
-- In Oedipus Rex, Oedipus's motivation for finding Laius's murderer is to lift the curse that is afflicting Thebes, but his determination to find the answer is can be understood an expression of moral courage and personal integrity -- or as a determination to understand the difficult truths about his own past. Either way, Oedipus's heroism has more to do with battling internal demons than defeating a physical enemy.
-- In Marie de France's Lanval, Lanval, the hero, is more subject than object: the fairy princess who loves him blesses him with riches, but only as long as he refuses to identify her. Here, the "heroic" moment for Lanval comes when, threatened with possible execution for homosexuality, Lanval breaks his oath by asserting that he does have a mistress far more beautiful than Guinevere. In so doing, he is at once breaking his word and asserting the superiority of his mistress, both an act of cowardice and a tribute to his lover. It is the fairy princess that arrives at Arthur's court in the nick of time to prove his innocence and save him.
-- In Sir Gawaine and the Green Knight, Gawaine is in a similar predicament: he is bound by honor to travel to the Green Chapel and submit to an axe blow from the terrifying Green Knight. In the face of certain death, Gawaine makes a difficult choice: he chooses to conceal from his host the green girdle of invincibility the host's wife has given him. This dishonorable act is ultimately pardoned by the Green Knight since his choice was motivated by a desire to save his own life.
These stories all have a few themes in common: illicit sexual relationships (Oedipus with his mother, Lanval with Guinevere, Gawaine with the wife of the host); the question of honor or integrity; and characters that are confronted with difficult choices that reveal internal weaknesses. In that sense you can read these characters as precursors to more modern conceptions of heroism.