Taking a pet in a rental car is not an issue as long as you look after some small details.
I researched many, many rental car companies, and could find no policies that restricted the transport of dogs, cats or any pets in rental cars. The biggest factor that you will need to be careful of, is that your pet does not dirty the vehicle (beyond what would be considered normal use).
You will need to make sure to clean the car thoroughly (vacuum out any pet hair and deodorize to eliminate any pet odors) or else you could incur a cleaning fee. While cleaning fees can vary by rental company, common fees for smoking in rental cars can run $250 or more.
You'll also want to make sure that the animal is protected properly in the rental car, either by special pet harnesses for cars or in a proper pet carrier.
Be sure your pet wears a collar with complete identification and a license tag. Have a rabies vaccination certificate if you will cross state or international borders.
You must also have a health certificate, especially if you intend to enter Canada or Mexico with the car rental.
If your pet is not used to driving in a car, start with small trips at home before heading off on a long road trip with your rental car.
Try talking to your pet if they seem nervous or pop in a cd that you play often at home. The familiarity of your voice and/or the music should be soothing. A favorite toy, blanket from home or stuffed animal will help as well.
One trick that we always find works well is to cover our dog's kennel with a blanket. (We do this anytime we crate her, not just when travelling). Most dogs enjoy a cave-like atmosphere and also, they will not be distracted visually during the trip.
Keeping your pet relaxed and stress-free will go a long way to making a good trip experience for all involved!
When taking a pet in a rental car, stick to your regular feeding routine, but don't give an extremely large meal until you've reached your destination. (This tip is courtesy of a car-sick border collie puppy and her new, inexperienced owner. I learned my lesson!)
Dry food is a lot easier to take in a car and you don't have to worry about trying to keep it refrigerated. Take along a plastic jug of cold water. If you can, plan to stop every two hours for a little exercise and remember to include a leash in your travel kit.