Will your insurance go up if someone else is driving it and gets in an accident?

The question is too general to properly answer. The issue your question seems to be really directeed to is the insured's liability for negligence. It is possible that you could be held responsible for the acts of another authorized to use your vehicle if the driver causes an accident that results in damage to persons or property; including your own vehicle. I suggest that you augment the question with fact specific information. In short, what happened? Which State did the accident happen? Did the accident occur in the same State your residence is and where you entered into the contract of insurance? What type of damage was there and have you or someone else made a claim for policy coverage. The answers to these questions are some of the things necessary to give you an accurate answer. They are examples of some of the questions for information that would be needed.

Tort law[civil wrong] and contract law may differ by state. The best answer to your question as posed is to speak with a lawyer of your choice in the area you live that you believe to be competent in the field of auto accidents and seek advice.

A helpful hint: most state and local bar associations have web sites and toll free telephone numbers for legal assistance for people who lack the means to hire a lawyer. The lawyers in the United States have Pro Bono programs set up in each state which means that if you cannot afford a lawyer's fee for consultation, lawyers will donate their time to help you. They also have referral programs with toll free numbers.