It was initially thought that Martians had a physiological vulnerability to fire. Now, after a desperate rewrite, it has become an implanted psychological vulnerability that was set up in order to keep them from enjoying their full potential as Burning Martians and being too powerful.
This makes Superman's heat vision much less of a factor.
So what does it come down to?
Common Traits:
Strength
Although Martian Manhunter can alter his strength to a degree, he would still only be within range of Superman's level, not equal to it. And he wouldn't be able to maintain such an adjustment indefinitely.
Superman would still have the advantage.
Speed
This is similar to the strength issue. Even augmented, Martian Manhunter only approaches Superman's level, but Superman is still superior.
Durability (physical)
Again, Superman comes out on top.
Flight
Speed may affect this, but as that's already been addressed, neither has a definitive edge here. Both are able to fly with excellent control.
Martian Manhunter's Additional Abilites:
Telepathy
This is probably the most significant factor in the fight against Superman, as it's the only area where he really has an advantage (and a huge one, at that).
Superman has the strongest will on the planet (JLA #52), but without any actual mental powers of his own, he'll eventually succumb to Martian Manhunter's assault.
Intangibility
I see this being put to better use defensively than offensively. If he is able to use this in conjunction with his telepathy, then it's pretty much over for Supes.
Superman's Additional Abilities:
Heat Vision
Although MM has overcome his psychological vulnerability to fire, that doesn't mean it won't hurt him. Superman's heat vision isn't a simple bonfire; as far as I'm aware, it's multiple times hotter than the surface of the sun, perhaps approaching, or even exceeding, the core.
What it comes down to is whether Martian Manhunter uses intangibility and telepathy together. If so, he wins after slowly crushing Superman's mind. If not, he loses after delivering some serious pain to the Man of Steel.