It would differ from time, place, and circumstances.

Courtesy of /u/Iphikrates, the ancient Greeks, at least in the Classical period, repatriated their war dead for public burial. The exception was the Spartans, who buried them on the battlefield. You can read about that here. You can also read about how the Persians buried their own and the Greek dead after Thermopylai, but after Plataia the Greeks left the Persian dead to rot here.

Tacitus describes Germanicus arriving at the scene of the battle of Teutoburg forest six years after the event thus:

Germanicus upon this was seized with an eager longing to pay the last honour to those soldiers and their general, while the whole army present was moved to compassion by the thought of their kinsfolk and friends, and, indeed, of the calamities of wars and the lot of mankind. Having sent on Cæcina in advance to reconnoitre the obscure forest-passes, and to raise bridges and causeways over watery swamps and treacherous plains, they visited the mournful scenes, with their horrible sights and associations. Varus's first camp with its wide circumference and the measurements of its central space clearly indicated the handiwork of three legions. Further on, the partially fallen rampart and the shallow fosse suggested the inference that it was a shattered remnant of the army which had there taken up a position. In the centre of the field were the whitening bones of men, as they had fled, or stood their ground, strewn everywhere or piled in heaps. Near, lay fragments of weapons and limbs of horses, and also human heads, prominently nailed to trunks of trees. In the adjacent groves were the barbarous altars, on which they had immolated tribunes and first-rank centurions. Some survivors of the disaster who had escaped from the battle or from captivity, described how this was the spot where the officers fell, how yonder the eagles were captured, where Varus was pierced by his first wound, where too by the stroke of his own ill-starred hand he found for himself death. They pointed out too the raised ground from which Arminius had harangued his army, the number of gibbets for the captives, the pits for the living, and how in his exultation he insulted the standards and eagles.
And so the Roman army now on the spot, six years after the disaster, in grief and anger, began to bury the bones of the three legions...

To give some examples from my area, Takeda Shingen was laying siege to Shiga castle in 1547 when the Kantō Kanrei sent an army to relief the castle. Shingen sent a detachment to face the relieve force, and the Takeda won. Afterwhich the Takeda took the heads of the 3000 dead from the relief force and laid them out for the Shiga castle garrison to see. In general though they would've been buried. After Sekigahara, for instance, Ieyasu paid 1000 koku to the samurai in charge of the area and had him take care of burying the dead and fixing any damaged temples. After the Siege of Ōsaka, a merchant camp follower by the name of Okamoto Saburōemon, known by his shop of Yodoya, volunteered to take care of burying the dead bodies, in return for the right to strip and sell the valuables he find. In Japan it seem to be custom to bury the dead and build at least a shrine for the dead and give offerings, like this one at Sekigahara, or at least a little plaque for the dead of a supposed 1467 battle, so the dead don't haunt you.

For the longest time, the people of Aizuwakamatsu held a grudge against the people of Kagoshima (Satsuma) and Yamaguchi (Chōshū) for invading them in 1868 during the Boshin War. One of the widespread stories is that the defeated locals wanted to bury their dead, but the government forbade it. This has been proven false in recent years, as documents came to light that after Aizu surrendered, the government ordered four Aizu samurai to organize the burial of 567 dead showing the dead were actually buried. In this case the victors had the defeated take care of the burial, and the legend show what the people thought should (and in this case, actually did) happen.

In chronological order:

/u/NMW has previously answered How were bodies disposed of in the First World War?

/u/CrossyNZ has previously answered How did the French clean up Verdun after WWI...? but naturally discusses the Imperial War Graves Commission.

Answers remain to be written about the actual French disposal of French bodies and German disposal of German bodies.

/u/eternalkerri has previously answered Who cleaned up the bodies on D-Day?

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