During the early middle ages, at least, one of the biggest war crimes was failing your lord. If you let him go into battle without you, or if he died while you fled, you would become a social outcast – at the worst an outlaw (as in, there was no law to prevent someone from murdering) and at the best a mercenary.

The poem Beowulf gives us both sides of this coin. Wiglaf complains on Beowulf's behalf as he is deserted by the companions he had favored with gifts. And after Beowulf's death, his former companions mourn their impending poverty exile.

War crimes as we think of them went largely unpunished. Charlemagne executed thousands of prisoners of war, barely raising a comment. And his biographers praised him when he enslaved his captives and redistributed them throughout his lands or sold them onward to English or Mediterranean markets. So I think we need to flip our concept of war crime to focus more on the demands of duty to one's allies rather than limitations of violence to one's foes.

I'm a little late to the party here, and "war crime" as a term did not come about until the 20th Century. There were, however, some acts against certain vulnerable populations that went too far and were broadly condemned. A great example of this is the event known as the "Harrying of the North," which I've written papers about. Basically, in 1069/70, William the Conqueror went into Northern England, which had risen yet again to support his opponents, and murdered thousands of civilians. He also destroyed farm land and induced a man-made famine that reduced large swaths of Northumbria to waste for a generation. Even in the Domesday Book, compiled about 15 years later, parts of the land was valued as worthless.

Norman Chroniclers, even those who usually favored William, could not forgive him for this. Illustrative of the sentiment is Orderic Vitalis, a monk:

The King stopped at nothing to hunt his enemies. He cut down many people and destroyed homes and land. Nowhere else had he shown such cruelty. This made a real change.

To his shame, William made no effort to control his fury, punishing the innocent with the guilty. He ordered that crops and herds, tools and food be burned to ashes. More than 100,000 people perished of starvation.

I have often praised William in this book, but I can say nothing good about this brutal slaughter. God will punish him.

I don't have my books on me for proper historical citations at the moment, but the [wiki article is actually quite accurate and detailed.] (https://en.wikipedia.org/wiki/Harrying_of_the_North)

That's about as close as you'll get to a "war crime" in medieval texts.