Whether or not your renters insurance would pay to repair or replace a garage door depends on the nature of the incident. If you or your immediate family are responsible for the damage, then your renters insurance might cover the repairs, but only if the cause was accidental. Before you file a claim, give some consideration to the deductible, the total cost of the damages, and how many liability claims you have filed in recent years.
One aspect of renters insurance is to cover damages you cause to the property. For example, if you accidentally broke a door window, your policy would cover the liability. If you backed your car into the door, the situation is more complicated, and the coverage will belong to either your auto or renters insurance policies. In that case, file on your auto insurance first, and fall back on the renter’s coverage if the insurance company denies the claim.
If the damages were not caused by you, then the damages should be the responsibility of the property owner. In this situation, the claim would be filed by the homeowner against their landlord insurance policy. Unless you caused the damages, you are not responsible for repairing the dwelling or other structures. The property owner is responsible for repairs related, for example, to wind storms, but may not be responsible for fire or water damage if you caused the damage to happen.
If the cost of repairs is only a little higher than your deductible, it may be more economical in the long run to pay for the repairs out of pocket. If you have filed any liability claims in the last 7 years, including home auto and other liability policies, filing a minor claim could be the final straw which dries up your premiums across all of your liability policies. If you aren't sure how many claims you have filed, check your CLUE report to see what information it contains.