The Spyro the Dragon games developed by Insomniac also had some Amusing Injuries too, but nowhere near as much as the Crash series.
Wario. Oh, Wario. Throughout the main series of games starring him he's been stunned, stung and inflated, ignited, flattened, compressed, zombified, vampirified, frozen solid and then shattered to no ill effects except being dazed or moving erratically for a while, that in exchange granted him the ability to solve numerous puzzles along the way.
The entire point of Sonic The Fighters (A.K.A. Sonic Championship), which was filled to the brim with Looney Tunes-style slapstick and injuries, and even though the game itself is not very liked, the slapstick of it sure is.
The first Sonic Riders had quite a lot of cartoon slapstick too. Players can be flattened, burnt to a crisp, electrocuted, etc.
In Sonic Generations, Classic Sonic will be flattened vertically if he gets caught between the truck and a wall in City Escape.
The Fallout series play with incredible injuries in combat pretty well. Exploding heads, huge chunks blowing up out of a BB shot, bodies shredded to pieces by machine guns. Lots of fun. Even worse, you can a take perk called "Bloody Mess" to make it even worse.
The 'amusing' comes in the first two games from the running commentary, both from the characters and the combat log leading to such gems after shooting a woman in the groin as 'She took it like a man, that is to say, it hurt'
In Fallout 3, one can get blown across town if they stand next to a car when it explodes. Set the car on fire and walk away. Ten seconds later, you're watching your sorry ass fly.
Ace Attorney: While the setting is not prone to them (being a series focusing on murders) at a few point in the 3rd game Godot throws his coffee at Phoenix and it is played for laughs. The fact that coffee is HOT never comes up the first time it happens. When it is commented upon Phoenix's thoughts reveal it is only hot enough for first-degree burns.
And of course there's Franziska von Karma's whip, which afflicts absolutely everybody through the second game with no ill effects, except when she whips Phoenix into unconsciousness with it.
These are actually a gameplay element in Sphinx And The Cursed Mummy. The Mummy of the title is already dead, so things that would normally kill him instead have... other effects. Being set on fire allows him to set other things on fire, getting smashed flat by Smashing Hallway Traps of Doom makes him paper-thin and lets him slip through bars, and getting put through shredders turns him into three separate mummies. Though he doesn't get hurt, he definitely doesn't like being set on fire if you watch his Idle Animation...
The UK edition of Official Playstation Magazine had a semi-regular column where an actual ER Nurse would analyze a couple of different Amusing Injuries from some games like Ready 2 Rumble and Disney's Tarzan and describe their Real Life consequences in precise clinical detail.
In Tales of Monkey Island: Launch of the Screaming Narwhal, the current captain of the Screaming Narwhal encourages you to try to roust him from the deck of the ship (thus becoming captain yourself) by any means necessary, because due to Flotsam Island's odd winds no one can leave, so it's become something of a game, and he has proven so skilled at beating all comers that no one's even tried for two years, and he's bored. Once you succeed, he congratulates you on your captaincy and cheerfully excuses himself to have the doctor see to his grievous internal injuries.
In Bubsy, every time you die it is in a cartoony, amusing way. Examples include splatting when you fall too far, only to stand up and walk off screen while your midsection accordions, putting on an olde-timey captain's hat, holding up a miniature flag and saluting when sinking into water, and shattering when hit by a gumball launched out of a gumball machine.
Ultimate Spider-Man "Don't worry, nothing's broken except my spine, and my ribs, and everything else."
The Legend of Zelda: The Wind Waker is full of these, thanks to the game's cartoony, slapstick-y style. Notable examples include Link flying 50 feet in the air with his rear on fire if he falls in lava, flying facefirst into the side of the Forsaken Fortress in a launched barrel early on in the game, and later repeating the same (sans barrel) with the Tower of the Gods.
Interestingly, The Legend of Zelda: Twilight Princess has at least one instance of unrealistic injury, despite the game going into such detail as to show Link melting and sinking beneath lava, struggling and screaming. If Link falls into a body of ice cold water in the Snowpeak region, he instantly becomes encased in a perfect block of ice, rather than suffering hypothermia and sinking.
Dwarf Fortress certainly qualifies. Even with its incredibly complex anatomy system, a lot of injuries that would normally kill someone in real life is pretty much harmless in the game. For example, losing a limb in adventure doesn't cause you to bleed to death, like you would expect. In fact, you just have to walk a while for the wound to heal by itself ! And you can get injured (or injure someone else) in pretty unique ways. This is probably one of the reason behind the game's popularity.
A Bilingual Bonus example from the slain SS in Wolfenstein 3D (and in the secret levels in Doom II''): "Mein Leben!" note translation—"My life!" Particularly funny (What? They're Nazis...) in Doom II because SS spawn in clusters of four to compensate for the much better weapons in Doom II''. Given the game's love for the Supershotgun and the Rocket Launcher, often you will hear multiple SS crying out "Mein Leben!" all in a bizarrely amusing chorus of death.
Diligent users of the Scan visor in Metroid Prime were treated to a story about the Space Pirates attempting to reverse-engineer the Morph Ball. Hilarity Ensues.
The Darkstalkers and Power Stone series are more cartoonish and humorous than Capcom's other fighting games, and less serious too.
Bayonetta can get flattened by ball-shaped objects, such as the Golem when it is in its ball form, or the final boss's ice balls (which takes away some of the seriousness from that boss battle). When this happens, she will lose a tiny bit of health, then humorously flap around on the ground before she returns to her normal 3D state.
Lammy from the Pa Rappa The Rapper series goes through quite a few Amusing Injuries in her spin-off title alone, such as getting her belt caught on a door and being flung through time, then creating an Impact Silhouette when she lands on a far-away island (this only applies to the NTSC version however). She also gets struck by lightning if she plays badly in Stage 6, and shocks herself from plugging in electrical equipment during a flashback, resulting in a Funny Afro. However, in the JPN/EUR versions, the aforementioned scene of her being flung through time and landing on an island didn't occur, and instead she slips on a Banana Peel, dies, and goes to Hell, and must win the song in order to be brought back to life, none of which was Played for Laughs.
Lollipop Chainsaw might count. Zombies bleed sparkles and rainbows upon being killed, and also, at the beginning of the game, Juliet has to behead her boyfriend Nick because he got bit by a zombie, and then keeps him alive as a talking head. And then, in Chapter 2, Juliet can get electrocuted cartoonishly by Vikke the Zombie Viking's lightning and electrical attacks, and she seems to have a cartoon skeleton despite looking somewhat realistic on the outside.