Filing a couple of small insurance claims will not cause your insurance rates to go up, but filing numerous claims against any combination of liability insurance policies probably will. Insurance companies expect their policy holders to file a claim now and then, or else insurance would be unnecessary. The only time those claims become a problem is when you file them too often, fraudulently, or under suspicious circumstances.
When you apply for any type of liability insurance, whether for your home, car, rental unit, or condo, the insurance company will request a copy of your Comprehensive Loss Underwriting Exchange (called CLUE) report. This report will detail all of the insurance claims you have filed within a given period of time, and can alert insurance companies to people who are filing an unusually high number of claims.
If your CLUE report indicates that you have filed an excessive number of claims, insurance companies can and will refuse to sell you a policy. What happens more often, though, is that the company will set your premiums to account for your high rate of incidents. So instead of being refused an insurance policy, you are more likely to be accepted for the coverage, but forced to pay higher premiums which balance the increased risk of insuring you.
To avoid having a CLUE report, don't file a claim for every little thing which happens. If, for instance, the damages are only a little higher than the deductible, go ahead and pay for the damages out of pocket. You will still have to pay the deductible next time, but your insurance policy will have fewer claims against it, and that will result in lower premiums the next time you have to buy a policy.
Similarly, do not allow your insurance policies to lapse or expire. If you know your car insurance is about to expire, go ahead and start shopping around for a new policy. You may choose to stay with the company you have or switch to a new company, both without being penalized. But if you allow a policy to lapse, then it will cost you more to get a new policy, and some insurance companies may even deny coverage based on the idea that you may not keep the policy in force once you get it, raising the risk of insuring everyone.