I find it really hard to exercise outside when it is over about 93-94F and humid. Basically when it gets up near body temperature outside and it's humid. I can do it, and I have, but I don't like it, and it can put me off of exercise for a couple of days.
Recently in my area it has been getting near 100F every day for most of this week and is very humid, and like you I don't want to either eat or exercise.
The downstairs of our house is cooler, so I'll go down there and exercise, which helps. There isn't much room so I can't do much, Burpees don't take much space so I usually do that.
If I don't feel like eating, I just eat light, and keep it Paleo, like for breakfast I might have a glass of iced coffee with a tablespoon of heavy cream, and that's it, then for lunch, a couple of eggs or a salad. I find a strict Paleo diet is a natural appetite suppressant, so eating the right foods keeps the munchies away. This is a good time to try to bend rules, and not break them, like I normally try to eat breakfast and avoid coffee, so iced coffee + cream is a small cheat, but hey it can't be 100F forever.
If you feel yourself starting to eye the Ben & Jerry's section at the supermarket, at least force yourself to eat a couple of eggs (or substitute your easy-to-eat fully-Paleo protein source here) first. Today in fact I had a handful of peanuts (a cheat for me, I tend to overdo it with them), for the same reasons... the heat gets me off track.