Sometimes slicks can provide so much grip cars can overturn without sliding, it is linked to the level of grip v centre of gravity. But you didn't ask about the risk, consequences or chances of this, you asked about damage through usage.
I think the answer is yes, there will be some. More of a wear and tear being increased. I have not run Clio's with slick tyres, but over time this has been done on other similar cars.
Things that have happened are; stub axles have failed/sheared. Wheel bearing life has shortened. Subframes have cracked. This tends to be an over time thing though. Forces being transmitted into driveshaft cv joints because the engine and box is thrown around more is true. You might get problems with gear selection as you'll run out of revs in different places on a lap, be watchful of mis shifts if running street spec engine mounts. Just occaisionally things like wiring and piping is forced into contact with things it is not meant to touch, heat damage can sometimes be a problem. This is a precis of most the increased load problems experienced when streety vehicles get race related grip. I don't think the diff will be damaged at all directly. It might be affected from excess lateral load being transmitted into it - its just possible the gearbox case could distort to affect bearing clearances and running temp, but unlikely. You have to get things very hot to go over what the oil would withstand.
By all means have fun, try it. But if you are running an increased pace and exploiting the additional grip its not without its wear and tear consequences.
It is worth doing some research into knowing how to look after true race tyres, and know when they are finished. Some slicks are treadless street carcass tyres - these would be about idiot proof... others are highly sensitive products that will fail if mis used.
Andy
