I thought it would be interesting to do a quick google search for "armor in the desert". Because, as always seems to be the case, people make a lot of assumptions.
The result? People in armor wore white cloth over their armor and drank a lot of water.
Which actually matches my experience (somewhat) with hiking in the desert. It may look odd that I wear a hat and long sleeve shirt, but it's frequently better. It's one of the reasons long flowing robes are so common in the middle east.
https://history.stackexchange.com/q...-use-of-mail-armor-in-the-deserts-of-the-midd
https://www.reddit.com/r/AskHistori..._did_heavily_armored_knights_handle_the_heat/
https://www.quora.com/How-could-med...ll-plate-armor-in-middle-east-during-crusades
It's also important to point out that most, if not all of the responses, were regarding chain/ringmail armour and a surcoat, rather than full plate - and excellent physical conditioning is something I hope we can assume on those knight's behalf!
Simply standing under a blazing sun shouldn't pose much of a threat to a heavily armoured adventure. Strenuous physical activity should prove to be harder, though as a character can, at least according to the rules without DM arbitration, perform a back-flip in something like plate armour as readily as if they were naked, perhaps this is a moot point. Combating the stress of marching/fighting hot environments seems to be a common theme with the articles scattered around good 'ole Google Scholar, and common sense tells us that its going to present more of a challenge to march/operate in such conditions when encased in metal plates than otherwise.
In the Sandstorm desert adventure guide, things really (incoming pun) heat up (pun deployed) around the 111 Fahrenheit/44 Celsius mark, and for our games at least, such environments are very rare outside of an actual Hollywood-inspired blazing desert. And here, dealing with the heat, the need for water and the night is all part of the fun.
So, with all these things considered, and heavily ripping of the Sandstorm book, I'd hand-wave any checks outside the extreme heat conditions. Sandstorm introduced a level of granularity by imposing subdual damage on those that failed a Con save after a time, and eventually, in the unearthly high temperature conditions, lethal damage.
Lacking subual damage and with 5E favouring simple though sometimes chunky mechanical systems, I'd set a straight Con save depending on the temperature, with heavy armour users making the check at disadvantage unless they were dressed/outfitted in 'Desert Clothing'. Those in other armours can make the check as per normal, with advantage if they have 'Desert Clothing'. Desert Clothing being a catch-all term for the various techniques employed by folks investing some resources into combating high temperatures. Seems simple enough.
