The cliche tells us, “it’s not the heat, it’s the humidity.” It’s my understanding that that’s because when the air is humid, sweat doesn’t evaporate readily, and we lose the cooling effects of perspiring.
Since fur-covered mammals don’t perspire, is there a significant difference for them in how unpleasant it is when it’s hot and humid vs. when it’s hot but not particularly humid?
I guess animals that cool their blood by sticking out their tongue and breathing through the mouth suffer as well, as they too rely on evaporative cooling. Even covering yourself with wet mud won’t cool you as efficiently if the air is near-saturated with water than if the humidity is lower.
I think a lot of animals that don’t sweat still rely on evaporative cooling. Dogs, for example, pant to rapidly move air over their big fat wet tongues; cats will do this too if they’re warm enough. Kangaroos lick/drool on their forearms. And various other animals are known to wallow in the mud. Any such animal will have a more difficiult time keeping cool if the relative humidity is high.
Actually most fur-covered mammals do perspire. Very, very few mammals have NO sweat glands ( whales among them ). Dogs and cats for example sweat through their feet and nose. On hot days you can sometimes see dogs leaving a trail of wet footprints.
My brother’s a fur covered mammal… and he’s not doing so great with the humidity… 
J/k… our hamster doesn’t do so well with the heat and humidity. She tends to rub her back against her water spout thing… it cools her down, pretty well, I’d guess.
Seals don’t seem to mind the humidity.
Okay, I got the “no sweat glands” thing wrong – but clearly sweating isn’t the main way a cat or dog stays cool. Does a dog pant more on a humid day than on a less-human day that’s the same temperature?
A recent scientific article says that humans did well evolutionarily partly because of our long term endurance in hot weather, which lets us run down much faster animals until they collapse with heat exhaustion. Our sweating was the primary adaptation we use for this. I think the gist of it was that, while some animals such as lions would hunt in the hot open plains of Africa by chasing down a gazelle (or whatever) at high speed, humans would do it at a much lower speed. While the gazelle evolved to run very fast for a little while, it needs a long recovery time to shed the excess heat after the lion caught someone just a bit slower - which wouldn’t have taken very long. A human chasing it for hours never gives it that chance, and can shed heat at a much faster rate with essentially all our surface area wet and exposed to the breeze.
This is oblique to the OP but is consistent with the view that exporting lots of heat primarily through sweating is quite a specialty of humans, something we do way better than any other species, or better than almost all.