If someone else gets a ticket while driving your car will your insurance go up?

If they were driving it with your knowledge or permission, yes.

Actually, no it won't.

Each person is liable for their own actions, regardless of the car they are in.

The driver's insurance will go up and would also be liable for any criminal acts.

I would assume that if someone else is involved in an accident, while driving your car, your rates will go up?

Your insurance will NOT be affected if someone used your car and got a citation BUT,,, If they had an accident and had NO insurance YOU would be involved. If your registration and or insurance is not up to date, you and the driver will pay some fines and get violation points for the above.