Driving a rental car across the border can be tricky. Whether or not a rental company will allow this really depend on a variety of factors, including the car rental company and which border you are trying to cross.
However, you may need additional insurance in order to drive in another country and you will need to find out if there are any differences in the road you are heading to.
You have a lot more leeway here than if you are wanting to cross the border into Mexico. The majority of car rentals in both Canada and the US will allow you to take your rental car across the border, as long as you are planning to come back again. The exception to this rule is if you are a resident of the country that you are driving into. For example, if you are a Canadian citizen, you can not take an American rental car into Canada without it has Canadian license plates.
Also, if you are going to be taking a rental car across the border into Canada, you will need to show your rental documents to the border officials, so have them handy. In addition, you will probably need to purchase additional insurance, since the insurance that comes with the car rental won't cover accidents or theft outside of the US You will probably need to have a Canadian non-resident insurance card in this case.
Talk to your rental company before trying to cross the border, since you don't want to be crossing without permission or insurance!
This is a bit trickier. You'll be hard pressed to find a car rental company that will allow you to take a rental car into Mexico, due to the high crime and theft rate in this country. There are a few rental companies that may allow you to take the rental car across the border, but this is fairly unlicuntarily. A better option would be to just park at the border and walk across. If you need to have a vehicle on the other side, rent from a Mexican car rental company as they are better equipped to deal with the risks.
You will definitely need to buy extra insurance if you do decide to cross the border into Mexico in a rental car and manage to get permission from your car rental company. Sometimes, you can get this extra insurance to cover you in Mexico from your rental company, but you may prefer to go with a third party insurance supplier if it is cheaper. Read the contract carefully and make sure that you are covered for everything from car theft to accidents. Mexico can get a bit wild, particularly in the border towns, so try to park the car in safe areas where you pay someone to keep an eye on it.
Taking a rental car across the border is possible in many cases, depending on which border, but do make sure to talk to your rental company first about your plans. They may require extra insurance or have some tips for you. You can also find out what is required in order to cross the border, whether or not you need a passport and what other paperwork is necessary.
As long as you do your research, you should be fine to take a car rental across the border. Just be sure to have everything in order, including the extra insurance, so you won't be faced with unexpected promises.