It depends.
Most leases DO NOT include fish under a no pets section. However, some leases consider fish to be a pet and do not allow tanks (rare, but it happens). There are also locations that charge a deposit for tanks over a certain gallon size, rather like a pet deposit for a dog or cat. Check with your landlord and/or leasing agent to find out exactly what they mean. Most leasing agents and landlords are quite understanding about fish since they don't scratch walls, pee on carpets, smell, etc, and many are quite alright with the idea of fish being there so long as they have prior warning.
(*coming from someone who has survived three completely different leases in regards to their pets policy- from two different apartment complexes and a private landlord)