The HOA is in breach of covenant, if it is enforcing a nonexistent rule. The HOA has also committed the tort of intentional interference with contract (i.e., your lease agreement), because it is preventing you from enjoying the benefits of your lease with the property owner. And, the HOA is committing a private nuisance, which occurs where a third party acts in a manner that deprives a person in lawful possession of property of the quiet enjoyment of their possession.
Unfortunately, there is no government agency to which you can complain. This is one of those circumstances, where you will have to demand that the HOA immediately activate your gate remote, or you will be forced to sue for damages and an injunction. You can also demand that your landlord get involved, or risk being sued as well, because if the landlord defends the HOA, then he is in breach of the lease agreement, too.
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