Not much, actually
To start off with, let's just see if there's any creatures with a lower core body temperature (I'm probably gonna use that term a lot, so I'm gonna acronym it CBT.) than humans. According to 'Biology, Medicine, and Surgery of Elephants', elephants have a CBT of 96.8-98.6F (or 36-37C for anyone not living in the backwards imperial system.) There you go, a creature with a lower CBT than ours...ish. Sloths are even lower, with a CBT of between 80 and 93 degrees, according to wikipedia. I also found this riveting article from the scholarly journal Functional Ecology that discusses this exact question. Apparently, mass significantly impacts the CBT of some families of animal. I'd read through the article 4 or 5 times just to digest the highly scholastic language. Also take a look at some of its references. They seemed to contain some interesting (though not necessarily useful) information.
Or take most animals that aren't either birds or mammals. Backyard lizards, for example. They're cold-blooded animals, which means that they don't keep a consistent internal temperature. Boom, just make sure your world doesn't ever get up to 98.6F.
On the other hand:
There's a few good reasons that our CBT is where it's at (or near enough, we fluctuate constantly.) This article, which appears to be written by a hardcore phi conspiracist, claims that a lot of the important temperatures in our body are phi away from the freezing and boiling points of water. Interesting, but without a reason why1, we can't change it.
Let's disregard that for a moment. Our CBT is actually pretty critical to a lot of processes. The chemistry of a lot of the enzymes used in our body works such that they work best at... 98.6F. Some of them work better at fever temperatures, but that's an extreme case with immunological reasons. Specifically, the reason we're not living at the same temperature we do when we are sick is that most of our enzymes don't work as well so hot.
Possible solution: Handwave some alternative evolution in there. All good.
There's a good reason that the CBT is higher than the surrounding air: Movement. 671
If we were isothermal with the environment, we'd overheat whenever we did... anything. (Fish and reptiles are different.2) Keeping ourselves comfortably above the environment temperature means that we can shunt off excess heat after strenuous exercise through sweating, etc.
What would you have to change?
To start with, you're not gonna have a human when you're done. Accept it and move on.
Solution 1: Lizardmen
According to numerous sources,dubious these already exist and they control the government. How would they work, though?
Cold blooded creatures stay at the same(ish) temperature as their surrounding environment. On the one hand, this means that they don't need to spend as much energy on maintaining a CBT. On the other... The metabolism of reptiles is a lot slower than that of mammals of equal size. Since they have to work over a greater range of temperatures, they can't work as efficiently, which means that they're not as active. Your lizardmen would move slowly, with quick bursts of speed.
Lizards spend a large portion of their day sunning themselves to increase their temperature. In part, this feels really good.3 More importantly, though, the increase in temperature speeds their metabolism and allows them to digest things faster.
Pros:
* Reduced resting metabolism: Because cold-blooded animals don't have to keep a stable temperature, they're able to survive much longer without food. A crocodile, for example, can survive six months without a meal.
* IR invisibility: Warm-blooded animals show up pretty bright on thermal cameras. In contrast, since cold-blooded animals are the same temperature as their surroundings, they're harder to detect on IR cameras.
* Longer (resting) anaerobic respiration: With a low metabolism comes a slow energy consumption. The internet places the lung capacity of a snake at 30 minutes of air.
Cons:
* Reduced speed and mobility: Most reptiles hunt with a 'sit-and-wait' strategy. Because they can't use energy as fast as mammals, they're not very good when it comes to running down their prey.
* Reduced cranial mass: Most cold blooded animals... Aren't that smart. (Remember that velociraptors were dinosaurs, which are in a grey area.) The reasoning behind this is fairly simple: Brains take energy to run, and lots of it. In fact, an estimated 20% of human energy consumption goes to the brain. 671