What exactly happens to leather armor in different environmental conditions, for example battles or long exposure to rain or desert heat? What can be done to negate or care for it?
I have one, and I did wear it with several weather (for medieval feast).
Rain and water :
Rain and water change its color (making it darker). It's not too bad when it's wet, but it's hard to dry it without destroying it. While drying, it becomes harder. If you bend it while it's wet and let it dry, it's very hard to have it soft again. It's all twisted. If you ruin the color, don't expect to have the same color again. You can easily make it darker, but with water (and sweat...) you will have marks everywhere (For a medieval armor, that's actually not bad)
In order to avoid the hardening, you have to oil it before rain, with linseed oil at best (it make it darker too), or anything fat enough (shoe polish is not bad). Then make it dry without heat (no flame, no warming, let it dry slowly), not bended, and oil it again.
Heat makes you sweat, and sweat is salty, and it's the worst. Heat itself don't ruin the leather. If it becomes too dry, grease it. But the sweat makes white and wet halos everywhere. Don't use water to remove the salt. Oil or grease, again.
Even a very old and maltreated leather can become something beautiful if you grease it enough !
Leather is the tanned version of skin, so to some extent, you can see the effects of weather on your own skin.
In hot weather, leather tends to dry and crack, which results in it being less durable and more prone to tearing, it would also become more stiff and rigid (thus rather uncomfortable to wear) as well as losing some of its waterproof properties. The best way to solve that is by moisturising it, the same way you would do with your hands. Depending on the time and setting of your world, there are several products that can be used. If it's set in the past, your people could use oils (whether of animal or plant nature) to rub into it. If this takes place in the future, you could be more liberal about the choice of product (they could still use natural oils, but also stuff like shoe polish or even specialised products).
Water is not good for leather either. Although leather does tend to be more waterproof than other fabrics, water will get through it eventually and it will, again, lead to a loss of durability. You can read a whole article about how to take care of leather in damp conditions here: http://www.leathertherapy.com/pages/Waterproofing-Leather-.html
As for the way it behaves during battle, ultimately that comes down to how much action it sees, whether you get hit by the enemy, whether there is fire involved as or even chemical weapons. At the end of the day, there isn't an exact number of weeks/months to say how long a weather armour will last for, because it depends greatly on the type of battle that is going on (whether it's a siege that lasts for months or even years, or just a skirmish, and so on).
What kind of leather armor? It is supple leather that is padded and worn rather like clothing? Is it thick leather that has been hardened and used like plate? After doing medieval reenactments for years I have experience with both.
The person's sweat can soak into supple leather and leave white marks anywhere it touches. Heat can make the wax hardened leather softer. Armor that is allowed to get damp and not dry out properly can grow mold. It will smell like the nastiest gym bag ever.
As the other commentators have mentioned cleaning, drying, and oiling leather equipment keep it in good shape, water resistant, and long lasting. In a long campaign or siege without the time or materials to do this properly you'll have straps breaking. Edges will curl and deform making the armor less comfortable. The surface will be rough up and not smooth. Sweat stains will obscure heraldic markings.