Landlords have the right to deny a lease to anyone as long as the reason is not that the person is a member of a "protected class"--ie they can't decide to not rent to someone because of their race, sex, religion, etc.
Gun ownership is not a protected class. It is therefore legal to not rent to somebody because they own guns.
I would think that a civil contract that contravenes an established constitutional right would not be enforceable.
TheConstitution only limits what government can do. It does not restrict private citizens or corporations. An employer can restrict your wearing of a shirt that says something that they do not like--if the government did that, they would be violating your First Amendment rights. Likewise, a landlord can restrict your ownership of firearms on their property; in many states though, a landlord who recieves government money (ie government-subsidised housing) cannot restrict firearm ownership by tenants.
I live in a rented apartment. My complex has no written policy on gun ownership that I can find, but I didn't ask. Asking gives them a chance to tell you know. If they did have a policy prohibiting guns, I would take my money elsewhere.