He died (see below).
Basically every source is contradictory on this one. The official novelisation states that he/they were simply stunned, however the new (and hence fully canon) Star Wars Junior novelisation gives the very strong impression that he/they were killed by Han.
He had to give them a little credit—both stormtroopers were quick reaching for their guns, but very few in the galaxy were faster on the draw than Han. The two stormtroopers fell just as hard as the scanner crew did. “Nice shooting,” Luke said, clapping Han on the shoulder.
A New Hope: the Princess, the Scoundrel, and the Farm Boy
This concurs with the older junior novelisation. Interestingly, the scanner crew weren't killed in this version, just knocked out.
The two stormtroopers glanced at each other, then marched up the
landing ramp. They had no idea that the scanning crew had already been
knocked out, and that it had been the Falcon’s captain who’d summoned
them into the ship.
Han fired his blaster pistol twice. The stormtroopers never knew what
Star Wars: A New Hope - Junior Novelisation
(Note that Han's blaster doesn't have a stun setting).
That being said, in the Star Wars: Fact File, it's noted that the guards were merely stunned.
The prospect of leaving his escape from the battle station in the
hands of an old man did nothing to improve Han Solo's temper. However,
the calm air of confidence Kenobi exuded soothed Solo's doubts.
Stunning the scanning crew and the two stormtroopers was not
difficult. The overconfident Imperials had left no other guards.
Official Star Wars: Fact File 37
But which conflicts with the Star Wars Radio Play
COMM VOICE: What's your situation there?
2ND TROOPER: Sir, we're in the command office overlooking the captured starship. The scanning crewmen on the ship were knocked
unconscious, and the guards have been shot.
COMM VOICE: Any sign of the Rebels?
2ND TROOPER: None, sir.
Which actually lines up with the early draft script.
GUARD: You have business here?
The guard is Han Solo. He struggles to hide a large blood stain under
his right arm.
OFFICER: I thought I heard something down here.
As to why the trooper armour doesn't have any blood/scorch marks, this was addressed by authorised Star Wars artist Chris Trevas in his iconic "The Folly of TK-421" picture, commissioned by Lucasfilm in 2015 as part of their annual Star Wars Celebration
With regard to the ultimate disposition of his body/their bodies, only the official novelisation seems to address this. Note that this source indicates that he was stunned rather than killed, but regardless he was found aboard the Millennium Falcon before they escaped.
Someone finally discovered the two unconscious guards tied in the service lockers on board the captured freighter. Both men remained
comatose despite all efforts to revive them.
Under the direction of several bickering officers, troopers carried
their two armorless comrades down the ramp and toward the nearest
Star Wars Episode IV: A New Hope - A Novel
Thus even if he didn't die from his wounds, it strongly suggests that he died on the Death Star when it exploded.