If Superman died on Earth, would his body decompose?
He gets his powers and invulnerability by metabolizing the light from our yellow sun. If he is no longer alive, then his body won't be metabolizing that energy. In Tow of Babel for example this is exploited.
I don't think there has ever been evidence to suggest that the yellow sun affects his cells directly regardless of if he is alive or dead.
So, if Superman was dead and not metabolizing, he would then not have his powers, so I would expect he would decompose in the normal way Kryptonians decompose.
Alright, let's say he dies (Kryptonian cancer).
I'm not going to talk about his soul, so let's assume that the soul isn't an agenda here.
The absorbed sunlight in his body would still power a great deal of his cells, so it would take him some time if not years, to actually die (that is, if DC doesn't find a way of saving him again).
His Kryptonian cells would slowly vaporise, so his body wouldn't have the same odor as of a human body. So in short, he would pose as a giant energy container waiting to burst.
Having all this energy in one spot (presuming he is still in one piece), he would be like a dwarf star imploding into a black hole (there are also other possibilities here, turn into a neutron star, a magneter, burst/explode). What would happen to Earth and the human kind?
However, if I'm not correct and his organs really somehow decompose. His body would emit an amount of thermal radiation (an effect known as Hawking radiation), in the end leading to evaporation of his body.
I know I'm kinda late to the party on this one, but if you're willing to accept an answer from the DCEU:
At the end of Batman v Superman: Dawn of Justice, Superman dies. When his corpse is later recovered during the events of Justice League, enough time has passed that it should have started to decompose, but instead it's in seemingly-perfect condition. Bruce Wayne even predicts that Superman's corpse won't have decayed (though I can't remember his reasoning).
From that evidence, he would decompose either much slower than a human corpse, or not at all.