There are countless examples, throughout ancient and medieval military history, of how 'overheating', on account of wearing heavy armor, had a catastrophic impact on how the battle turned.
Some salient examples include the Battle of Carrhae, where Crassus was defeated and killed by the Parthians, as related in Plutarch's Life of Crassus. Here's a brief excerpt:
But the Gauls were distressed above all things by the heat and their thirst, to both of which they were unused; and most of their horses had perished by being driven against the long spears. They were therefore compelled to retire upon the men-at‑arms, taking with them Publius, who was severely wounded. And seeing a sandy hillock near by, they all retired to it, and fastened their horses in the centre; then locking their shields together on the outside, they thought they could more easily defend themselves against the Barbarians.
- Plutarch, The Life of Crasus
Several Crusader battles hinged on 'overheating'. According to the Arab chronicler, Ibn al-Athir, who wrote about the Crusades from an Arab point of view (more about him here: http://www.oxfordislamicstudies.com/article/opr/t125/e921), the Franks (or Frangi) lost the Battle of Hattin because by the time it began, they were already in agony, half-mad from the sun and tormented by thirst.
Anyone who has ever been through Hell Week or Basic Training, has experienced the agony of carrying a heavy load, while engaged in some frantic activity. Football players have been known to die after twenty minutes of such exertion, on a particularly hot afternoon, while dehydrated. The Crusaders who fought the Battle of Hattin had been suffering for two days.
A third example is Alexander the Great's infamous march through the Gedrosian desert, as related by Livius. His hitherto undefeated army was broken by the scorching heat, in what most historians agree was his greatest blunder. Here is a particularly relevant quotation from Livius:
[6.25.3] So there was nothing for it but to leave the sick by the way, and any man rendered incapable by exhaustion or thirst or sunstroke. No one could give them a helping hand; no one could stay behind to ease their sufferings, for the essential thing was to get on with all possible speed, and the effort to save the army as a whole inevitably took precedence over the suffering of individual men. - Livius
Anyone who has ever played a game of football on a hot day, will know how the steam of your own sweat inside the helmet can become intolerable. The first thing that most every player does, once they reach the sidelines, is remove their helmet and often douse their head with water.
It's not uncommon for concert attendees to die of heatstroke in the summer. Imagine if they were also marching and fighting in heavy armor. Whenever I attended an SCA battle, they had an emergency team there for people who passed out. Medieval warriors didn't benefit from having trained paramedics on hand who could safely remove them from the field.
So a lot of this is common sense. Unfortunately, they had no realistic way to prevent overheating, just as we still don't today. You can only minimize it by making good strategic decisions. If your army was lead by a competent general, then access to plenty of water, and rotations to allow some soldiers time to retire to the rear and eat/relieve themselves was just part of being a serviceable general. Both Crassus and Guy de Lusignan failed in outstanding fashion to situate their armies in such a way as to maximize their survivability.
But even if you had a competent general, more often than not, the task of the soldier was simply to suffer in place, as detailed by John Keegan, in The Face of Battle. He wrote that even though a majority of the English soldiers at Agincourt had dysentery (after a series of grueling forced marches), they were so outnumbered that they had no choice but to remain in formation, wearing their full panoply, prepared for a French attack. This meant standing in place for hours, while experiencing bloody diarrhea.
In conclusion: All historical evidence suggests that casualties to sunstroke, disease, starvation, etc, depended on how logistically competent (or unlucky) your general was. They dealt with it back then, the same way we do now. By dying, or discarding their armor, which also usually invited death. This is why many armies in history disintegrated without ever reaching battle. As soon as it became logistically intolerable, the men simply dispersed. If you have already discarded your weapon, shield or breastplate to survive, chances are, you won't stick around for the battle, unless you are trapped (as in the Battles of Hattin and Carrhae).