A Comprehensive claim might cause your car insurance rates to rise. It will depend upon state laws and if they do not dictate whether your premium can go up due to a Comprehensive claim on your auto insurance then it will be up to the guidelines and rating system of your insurance provider.
Some states do not allow for rates to be raised due to not at fault incidents, like a hail damage claim, while others states do allow it and then it would be up to your own individual insurance carrier's rating system to determine if your premiums would rise due to this claim. Many carriers now offer accident forgiveness, where the first claim and/or minor claims (like a Comprehensive claim) don't affect rates.
For example surcharges are specifically not permitted in New York for some items including claims made under your Comprehensive coverage. Your state's insurance regulator can tell you what your state's insurance laws are regarding making a Comprehensive claim and if an insurance company can surcharge or raise your rates for this type of claim.
All that being said, typically when you have a Comprehensive claim, your insurance carrier recognizes that it is due to something that is not your fault, and you had no control over, (i.e. theft, vandalism, hail damage, etc.) and do not raise rates. Since insurance carriers guidelines do vary to find out from your specific company if your rates will rise and if so by how much if you place a Comprehensive claim contact your agent.
Your agent should be able to explain your provider's rating system to you. It could be that if you make one Comprehensive claim it will not cause your rates to go up but if you end up making a lot of claims against your policy then your claims activity could cause an overall rating increase.
If your rates do go up, it may be time to start shopping around for a new insurance provider. You can start car insurance shopping here with us.