Not a lot of racing games have damage, but Need for Speed: Porche Unlimited did.

It had a career/campaign mode you could actually LOSE if you sucked too badly. If your car was too wrecked to win a race, and you didn't have enough cash to repair it, you're SOL.

Here's a sort of strange example: Metal Saga on the PlayStation 2 (no relation to the SaGa series).

In this game, much of the time you will be fighting in vehicles (though it's possible, and sometimes necessary, to fight on foot). While in a vehicle, when you take damage, your vehicle will lose armor until it reaches 0. (You can buy more armor in a garage, but that's the only way to restore it.) If your vehicle has no armor, the hit will damage a random part of the vehicle, and if the same part gets damaged twice, that part will be disabled. This can result in getting fewer attacks, or it can even make the vehicle unusable, forcing its driver out to fight on foot.

Now, it turns out that, later in the game, it's actually easiest to go without armor on the vehicles, as armor slows the vehicle down. By eschewing armor and choosing to use only weak but lightweight weapons, you get more turns, and if you are at a high enough level to compensate for the weapon's lower power (the formula seems to be additive), you will do damage faster. If your vehicle does take damage, you can repair it in the field, whereas you can't replace armor in the field.

Some games like Battletech, Kenshi, and Wresting Mpire 2008 deal with limb loss. A lost arm impacts your fighting ability, but a lost leg is generally more serious. In Battletech its an instant KO for the rest of the mission. In Kenshi losing a leg reduces you to a crawl until replaced by a mechanical prosthetic. In Wrestling Mpire 2008 the character can still stand up on their bloody stumps, but are generally unable to maintain their vertical base for more than a few seconds at a time. At least you'll get the opportunity to answer that age old question - how well can a one-legged man do in a butt kicking contest?

I heard that Metal Gear Solid games have such mechanic. Does anybody have knowledge about that?

In Arcanum an injury lowers your stats (depending on the injury)

In LISA: The Painful RPG the injury eventually prevents you from doing certain attacks

just 3. also when you lose your one eye your vision gets obstructed in first person view through the rest of the game.

I see OP started from Evil Islands and Betrayal at Krondor and Deus Ex were already mentioned...
I think I remember UnReal World keeping track of injuries on each body part and having effects for them, right?
And would the "wounds" in Lords of Xulima also count? It's not HP, you will fight the same at 100% or 1% HP, but some attacks cause wounds, and stats are reduced according to the number of wounds.

Darklands has a simple system like that: as your characters' stamina and strength get reduced by enemy attacks, they become overburdened and act much slower. The game doesn't distinguish between different body parts, though.

Not an RPG, but strategy: in Civilization (at least IV and V), remaining health impacts the combat strength of a unit. Japan's unique ability in Civilization V consists in ignoring the negative effect of injuries.