Lets say there is a person who wants to rent his car out to his friend
for a month In the United States and In Pennsylvania. He calls up the
insurance company and lets them know his friend should be driving the
car for a month. He then collects money for rent each month from his
friend. He does this for several months. Then the friend gets in a car
Was it insurance fraud to rent the car out to the friend?
No. If you told the insurance company what was going on, by definition, you did not defraud them. Fraud means lying about something or concealing something from someone. Nobody did that.
But, you may have violated the state law that requires you to have car insurance in force on the people driving your car up to certain dollar amounts, because you don't say what happened after the call with the insurance company to let them know this was being done.
Who is held responsible for the damages caused in the crash?
The driver of the car is always responsible for the damages caused in the crash by the driver's negligence (except in cases covered by Pennsylvania's "choice" no fault system discussed below). The injured person has to sue the driver not the insurance company (sometimes an owner of a car will also be responsible, but almost never an insurance company, except in the rare cases where the driver is insured but dies before being sued and has a closed probate estate).
The question is whether the driver is insured or uninsured for liability that is primarily the driver's responsibility, and if the driver is uninsured, what impact that has on the owner of the car.
Of course, if the accident is another driver's fault, your injured renter and companions can sue the at fault driver, and you can sue the at fault driver for the damage to the car that you owned (subject to no fault insurance considerations discussed at the end of this answer). You and/or the driver might get a ticket for being uninsured if either of you are uninsured, however.
The owner of the vehicle is, at a minimum, responsible for making sure that the vehicle and the driver have the state required insurance policies in place covering the vehicle and the driver.
Normally, if you contact the insurance company in advance, they would modify the insurance policy to provide additional coverage to this rental driver in this vehicle for an additional charge (which you can pay for with part of the money paid to rent the vehicle). The policy would have to be modified because usually a non-commercial automobile insurance policy will expressly exclude from coverage insurance on driver's who are renting your vehicles. This is what should happen.
For example, if your car is part of a car sharing program, usually it is necessary to get special additional insurance policies to do this.
Normally, people who rent cars can allow the driver to waive insurance from them if the driver has car insurance of the driver's own that covers their limited use of a rental car instead of the car car that the driver owns and usually uses (which is common, but is sometimes not a part of a bare bones car insurance policy), and if they also sign a form waiving coverage (which is usually required by state law).
If you rent your car to someone without having insurance coverage in place on the rental driver from either you or the driver, then you are violating state law by having an uninsured motorist. This violation, in addition to being punishable with a fine and maybe even a comparatively minor criminal incarceration sentence, also would usually make the car owner vicariously liable for any liability that the driver incurs in an accident (which would not be covered by the car owner's insurance policy).
Pennsylvania recognizes a cause of action for negligent entrustment (see also here) of your car in the hands of someone who owns the vehicle who you have reason to know is likely to get into an accidents (e.g. the driver has a suspended license or a history of drunk driving), but does not have general car own liability.
If both the car owner ands the driver have insurance that covers the accident (because the car owner's insurance covers accidents involving a rental driver), then the car owner's insurance pays first and the renter's insurance pays second if the car owner's insurance limits are exhausted.
If the driver did have insurance in an amount required by law and the owner did not have insurance in place, and the limits of liability were exceeded, the driver would still definitely be liable for the excess liability, but the owner would not.
All of the above is predicated on the assumption of a "fault based" insurance system, but Pennsylvania has a hybrid system that is a mix of fault based and no fault car insurance.
Pennsylvania is a "choice" no-fault car insurance state. That means,
if you choose no-fault coverage when you purchase a car insurance
policy, after a car accident you typically need to file a claim under
your own personal injury protection coverage to get compensation for
medical bills and other financial losses, regardless of who caused the
crash. Only if your injuries are serious enough can you step outside
of no-fault and bring a claim directly against the at-fault driver.
If the driver has no fault insurance and is insured, the driver goes to the driver's own insurance company for the driver's injuries and damages to the car, rather than suing the at fault driver, in a minor accident. Likewise, if the driver hits someone else and is at fault, but the person hit has no fault automobile insurance, the driver and the car owner won't be sued.