Should I file a claim if I damaged my car backing out of the driveway but I only have liability?

I ran over a chair while leaving the garage and my bumper got ripped off. I've never been in an accident or filed a claim.

If you only have liability coverage then you wouldn't be able to file a damage claim. Your policy would had to have included collision coverage at the time of the incident for you to be able to get it repaired through your insurance provider. Unfortunately you would not be able to add coverage after the fact for the purpose of filing a repair claim.