It normally does make your car insurance more expensive or less expensive if you lease a car compare to having bought the car outright without a lease or loan on it. Having financed a vehicle and thus a lien holder on the vehicle is not a rating factor for auto insurance companies.
Insurance companies may ask you during the application process if you have a loan or lease on the vehicle though to try and determine what types of car insurance coverages you need for the vehicle. Lease vehicles typically require higher auto insurance limits. The usual requirements for leased vehicles are Bodily Injury Liability of $100,000 per person and $300,000 per occurrence (100/300) and $50,000 of Property Damage Liability. Also physical damage coverages of Comprehensive and Collision are required with a deductible of no more than $500.
Since higher Liability limits then you may have decided to purchase if you owned the car on your own plus Comprehensive and Collision are required to satisfy the lessor this may make purchasing an auto insurance policy for your leased car more expensive for you. In this context some may view that a leased vehicle costs more for auto insurance then a car you own without a lien holder.
So if you own the car on your own you can decide on what car insurance coverages you want to obtain and maintain over the state minimum requirements while with a leased car your lessor will require you to carry certain limits and coverages.
If your car is financed with a loan or lease instead of owed in the free and clear by you is not a rating factor but the question may be asked by an insurance company because some carriers have filed rates with the state that required physical damage on financed or leased vehicles. Esurance is one example of an insurance company that has this requirement.
Many insurance companies ask if you own your car or if it is financed on the insurance application due to how they have filed their rates and/or to make sure that if the car is financed you carry Physical Damage since it is most likely required by your lien holder. If you do not put on the coverages that your lessor mandates, you will have issues with them.