The biggest two examples being the Eldar and the Tau. The Eldar are psychic space elves that can see the future. They fight against Chaos, which is what Eldritch Abominations fear when they're up late at night. The latest editions of the game has played up the "Manipulative Bastard" part to 11 and making it clear that they're all about the long game. In one example, they caused an Ork warlord to attack the Human world of Armageddon, killing billions in order to save a few thousand Eldar lives; in another, they happily purged of all life both Orkish and Imperium worlds in order to deny the Tyranids biomass. Further, an Eldar captured by the Imperial Guard claims that once they are back on top they will systematically kill every single human in existence (though this isn't a sentiment shared by all Eldar Craftworlds).
The Tau are idealistic space-communists that fight for The Greater Good, and are the closest thing the setting has to a neutral or even good faction. That is, if you ignore the fact that their entire race is being mind controlled by their leaders. Not wanting such an obvious good race in the series, Games-Workshop pointed out that the Tau's plans for humanity include forced sterility, labor camps, and genocide. The kicker? They're still the only thing close to a good race the game has.
The Imperium gets this too. For every hardworking adept who agonizes over every difficult decision, honorable space marine, or working class guardsmen with balls of steel they show you; they are contractually obligated to show ten Knight Templar inquisitors screaming "EXTERMINATUS!"
This is sort of built into the alignment system in Dungeons & Dragons, though it's more a result of any Evil alignment taking on Order Versus Chaos: chaotic evil and lawful evil characters theoretically hate each other as much as, say, good and evil ones, and while this isn't applied so much to mortals, the war between demons and devils, which is known as the Blood War, is mentioned more often than the war between celestials and fiends. It was a major element of the 2e setting Planescape.
The Third Edition Dungeon Master's Guide says "...evil rarely gets along with evil, for the desires of one selfish and destructive being, by definition, conflict with the desires of other selfish and destructive beings.
The two Fiendish Codexes explain that there is an infinite number of demons, and there's more of them spawning at all times. Angels, Archons, Devils & company are in a finite number. The Devils exist in fact so they can use their superior tactics and team work (due to their Lawfulness) along with similar weapons as those used by the demons, to keep the demons in check. The books make it explicit that if the Devils weren't around, the demons would swarm and destroy all of creation. The books also hint that if the demons vanished, the Devils could probably conquer the Multiverse, being a race composed entirely of Magnificent Bastards and Chessmasters. And in the 2nd Edition of the game, the various forces of good took time to fan the flames, hoping to break their enemies against each other. In the Planescape boxed set "Hellbound: the Blood War", there's even an adventure where player characters discover high-ranking angelic beings funneling weapons and armor to their favored side in the war so more and more of the fiends would be killed. Some demon sub-races are also enemies of other demon sub-races (bebiliths eat other demons, for example). Some members of the higher ranking devil sub-races get promoted to the next higher-ranking sub-race by getting their direct superior killed or demoted, while some pit fiends (the highest-ranking sub-race) do the same to replace the devil dukes and duchesses who themselves are The Starscream to the archdevils. There's also much enmity between the archdevils (Dispater and Mephistipholes vs. Baalzebul, Prince Levistus vs. Princess Glasya, all the other archdevils trying to take Asmodeous' throne) and the demon lords (Juiblex vs. Zugtmoy, Baphomet vs. Yenoghu, the three-way battle between Graz'zt, Orcus and Demogorgan). In the Complete Scoundrel supplement, a prestige class called the Malconvoker is introduced which follows the teachings of a book called Vital Pact to impersonate evil for the sole purpose of summoning fiends to fight other evils and other with perpetuate all of the fighting between evil above, since the Celestial beings are too few to win their war against evil otherwise.
4th Edition has several instances, the most notable being the god of war Bane and his eternal war versus god of destruction Gruumsh. While Bane revels in conquest and power, one of his prime commandments is to ALWAYS obey the rules of war. He also likes his followers to be rigidly disciplined and wants to preserve the world so that it will be worth conquering. Gruumsh on the other hand revels in utter destruction and encourages savagery in his followers. Both want to be the uncontested god of war, and thus they've been fighting for eons over their differences.
4th Edition retained the Blood War, but in heavily modified form. Firstly, its importance has been de-emphasized, citing its Plot Tumor-esque nature and the way it made Good and Neutral alike seem pointless and impotent. Secondly, the underlying motives have been changed from the lofty, philosophical abstracts to something much simpler: Asmodeus stole a shard of the literal Heart of the Abyss to create his infamous Ruby Wand, before he murdered his divine master. Asmodeus wants the rest of the Heart, figuring it would make him even more powerful. The other Arch-Devils want the Heart for themselves, figuring it would make them powerful enough to dethrone Asmodeus and conquer The Multiverse in his place. The Demons want the Heart's shard back. Like all of 4th edition, this change has been controversial.
An orc tribe's worst enemy tends to not be a human kingdom, a clan of dwarves or even a Hidden Elf Village, but a rival orc tribe, and the same applies to hobgoblins and their tribes. Only a sufficiently powerful Big Bad can terrify rival tribes into working together, and the result tends to be a Keystone Army. If the Big Bad dies, rival orc and hobgoblin tribes who were previously intimidated into cooperating will turn on each other with a vengeance.
The Chromatic Dragons (black, blue, green, red, and white), are all evil (blue and green are Lawful Evil, black red and white are Chaotic Evil), and all obsessed with gathering hoards, gaining territory, building power, and destroying anything good (especially the good Metallic Dragons). However, they have no love for each other either, and they will fight to kill or chase each other away (even if two dragons, a black and a green for instance, manage to form a truce, eventually and inevitably one or the other will break the truce and resume hostilities). The worst are red dragons, the archetypal evil dragons, who will slaughter anyone and anything in their pursuit of their desires.
Frost giants (Often Chaotic Evil according to the Third Edition Monster Manual) often kill white dragons for food and armour or capture them to use as guards. There are both many evil and many non-evil githzerai, but both attack the githyanki and mind flayers on sight. Meanwhile, githyanki and mind flayers are both evil and attack each other on sight as well.
Salamanders hate efreet.
Each beholder believes its appearance to be the ideal of the beholder form. As such, all other beholders are inferior at best, and ones that differentiate too much must be destroyed.
Though both races are Chaotic Evil, there's no love lost between the Qlippoth and the Demons; the former used to rule the Abyss before the Demons stole their place, and they fight ceaselessly to take it back or to hold it, respectively.
The adventure "Valley of the Brain Collectors", in the Iron Gods adventure path, has a struggle between the pseudo-Lovecraftian Dark Tapestry and its brain-eating neh-thalggu, and the actuallyLovecraftian Mi-Go and their brain cylinders, either for the brains or because the setting is running out of room for eldritch horrors and they've started jockeying for space.
Any two Ravenloft darklords whose domains share a border are quite likely to be bitter enemies, and most domains have one or more lesser villains waiting in the wings to seize power if the dominant evil should slip up. The most famous of rivals, Strahd and Azalin, have been feuding and sabotaging each other's schemes for centuries, although the nature of the Land of Mists prevents opposing darklords from simply overrunning their rivals' territory.
The post-war Dragonlance setting is like this. With Takhisis banished to the Abyss, her five Dragonarmies turn on each other. Now, the five factions are just as apt to fight each other as they are to attack the good guys.
The Character Alignment system for Palladium games (such as Rifts) establishes that Aberrant characters (think Lawful Evil) refuse to have anything to do with the other Evil alignments and finds them disgusting. It's quite possible to have an all-Aberrant party run a lot like an all-Principled party with only a few slight differences.
An In-Universe case for Rifts is the Coalition States vs. all D-Bees, once one reads the various sourcebooks and is forced to admit that the vast majority of non-humans on Earth genuinely are as evil and dangerous as the Coalition States portray them.
In the Old World of DarknessWerewolf, agents of the Wyrm don't take kindly to someone mixing in traces of the Weaver with their plans. This is likely to cause some mayhem when Pentex figures out the true backing of the head of their Avalon toy company subsidiary.
Savage Worlds: Necessary Evil envisions a world where supervillains are the only powerful humans left after a massive alien invasion. They form a resistance to fight back the occupiers, but aren't necessarily reformed. If they drive back the aliens, they'll be in a position to take over the planet.
An option for In Nomine casts the setting as Dark Low Contrast, where angels and demons are both wicked, violent fanatics and nobody at all is looking out for humanity's interests. The PCs in such a campaign will normally try to be better than the rest, though.
Crops up from time to time in Magic: The Gathering. In the Innistrad cycle, one subplot follows Liliana Vess, whose only goal in the plane is to kill the demon Griselbrand to keep him from collecting on the bargain she made with him long ago. Her pursuit of this agenda ends up saving the entire human population of Innistrad.
Pretty much the point of a game of Paranoia, as the characters, PC and NPC alike, are all backstabbing traitors.
Though it tends towards Grey and Gray Morality on the whole, Eberron has its share of purely malevolent elder evils. That does not mean said elder evils get along. The Lords of Dust are prone to internecine conflict, since different factions of them serve different Overlords who tend to have wildly differing agendas, and they all view the Inspired as a bunch of posers butting in on their turf when they have no right to; the Inspired, for their part, don't like the Lords of Dust because they view them (and especially the threat of them managing to wake one of the Overlords up) as a threat to their plans to create perfect order in the mortal world and thereby anchor their own dimension in its current Lawful Evil state; to top if off, nobody likes the Daelkyr, mostly because even the most powerful demon or Inspired can't predict what thosemaniacs want, and if the Daelkyr have any opinion on the matter (or any particular conflicts among themselves) they're not chatty about it.
The outright villainous factions in Mage: The Awakening usually dislike each other at least as much as the morally grey-ish Pentacle Orders. The Seers of the Throne are under the tyranny of the Exarchs, godlike ascended humans who seek to subjugate the world, and each of the ten orders is jockeying for favour and advancement at the expense of the others. The Scelesti, knowingly or unknowingly, serve Eldritch Abominations who want to corrupt and destroy the world, and have nothing in the way of large-scale coordination. And the one thing all of them can agree on is that the Banishers, who go insane upon gaining their magic and believe that Mages all need to die, can never be allowed to gain true power. And then there are the shadier factions with the Pentacle...
Iron Kingdoms has dragons who are, more or less by definition, arch-enemies. The conflict between Cryx and the rest of western Immoren has nothing to do with any actual hostility; it's because Toruk, the Dragonfather, wants to kill and reabsorb his wayward scions, and that's going to take a lot of resources. For his part, Everblight - the other dragon to have reached army-fielding status - is sufficiently not-much-better that his name is Everblight. There was also a historical Evil v. Evil conflict during the time of the malevolent Orgoth empire, the one time it tried to extend its reach into Cryx; Toruk sank their entire fleet in person, and from then until the rebellion drove them from Immoren, the Orgoth were very careful about the claimed borders of the Dragonfather's territory.