Like many warm-blooded mammals, primates have sweat glands all over their bodies, so they do sweat. When monkeys sweat, the water on their bodies evaporates to help them release heat and cool off.
Some mammals, such as cats and dogs, only have sweat glands on their feet, so they also pant to help them cool off. Whales are mammals, but they do not have sweat glands at all. Because they live underwater, they don't really need them to cool off. Very large mammals often have difficulty cooling off. As an adaptive mechanism, elephants have large thin ears that release heat quickly to help their bodies cool down.