That depends. Liability protection that you carry for personal injury and property damage will provide some protection while you are driving the rental car.
Damage to the rental car would be covered under Collision and Comprehensive Coverage if your policy has it. The rental car company may also try to recover damages for lost income while the rental car is out of service and diminished value. Your auto policy may respond to the loss of use, normally it does not. Your current auto policy does not afford coverage for diminished value.
Credit card companies often provide protection against these kinds of rental car claims so you should check there to see what the provisions and restrictions might be.
Finally, you can purchase a Collision Damage Waiver - CDW - from the rental car company. This isn't actually insurance but a release from financial liability you might otherwise be charged with as a result of damaging the rental car. The CDW is expensive at $8 to $12 a day. This would amount to over $4,000 a year for very limited coverage BUT any claim on this coverage will not affect your current auto policy or cause a surcharge. Still, if you do not have protection via your auto policy or credit card, paying the CDW over a few days may be preferable than being personally accountable for $15,000 or $20,000 or more to replace the rental car.