Yes—the fine print on your car insurance policy says you must inform your insurance provider of any address change. Failing to do so is a breach of contract.
If you don’t tell your provider your new address and get into a car accident, your insurance company could refuse coverage. As an uninsured motorist, you'd be solely responsible for any medical payments, repair bills, and damages you caused.
It’s not worth the risk. Tell your insurance company about your new digs and ask if they have any car insurance discounts that could lower your rates. Or you could always shop around for a better deal in your new city.
Congratulations on your move—make sure to pack your insurance paperwork in a convenient spot!