Many credit cards in the United States offer some kind of rental car protection (mirror). In case of a damage to the rented car, does using the credit card's rental car protection in case of a damage with car rental result in increase of some premium (as a typical car insurance contract would) or extra fees for the customer?
That would depend on your specific contract - they are not all created equal.
Typically, no negative consequences should arise; worst case, if you use it multiple times, they smell fraud, and will check each case very thoroughly, and finally terminate your credit card.
[I have used it once, and it was a smooth ride, with no consequences]