The federal Fair Housing Act requires that landlords make "reasonable accommodations" for tenants with disabilities. Allowing an animal can be such an accommodation (see the Fair Housing Amendments Act, 42 U.S.C. §§ 3601–3619.) State statutes may offer additional protections.
As a practical matter, assistance dogs are normally so well trained and well behaved that a landlord has little reason to object to them. The law allows landlords to include reasonable regulations in the lease or rental agreement. The owners, like all dog owners, are liable for any damage the dogs cause.
Still, landlords sometimes try to impose a policy of keeping out all animals, even those specifically allowed by law. One company, which owned an apartment complex in Washington, DC, told prospective renters—who were actually testers from the Department of Justice—that no dogs, including guide dogs, were allowed. The company ended up paying $25,000 to compensate would-be tenants who had been discriminated against and another $20,000 in penalties to the government. (See United States v. Douglass Management, Inc., consent order, 2006.)
Most courts have ruled that dogs do not need to meet the strict definition of a "service animal" as defined by the Americans With Disabilities Act; the FHA uses a broader standard. That means that even a dog that isn't specifically trained to help its owner might be a necessary accommodation.
As one court put it: "There is a difference between not requiring the owner of a movie theater to allow a customer to bring her emotional support dog, which is not a service animal, into the theater to watch a two-hour movie, an ADA-type issue, on one hand, and permitting the provider of housing to refuse to allow a renter to keep such an animal in her apartment in order to provide emotional support to her and to assist her to cope with her depression, an FHA-type issue, on the other." (See Fair Housing of the Dakotas, Inc. v. Goldmark Property Management, Inc., 778 F. Supp. 2d 1028 (D.N.D. 2011).)
People with disabilities who live in government-subsidized housing are allowed by law to have dogs, whether or not the dogs have any special training. HUD requires landlords to act in accordance with the federal Fair Housing Act and the Americans with Disabilities Act, both of which require landlords to provide reasonable accommodations to people who have service dogs.
If you want more information on the law, a good resource is Best Friends for Life, a booklet that explains the rights of people with disabilities (and others) to have animals in rental housing. It's available from the Doris Day Animal League and the Massachusetts Society for the Prevention of Cruelty to Animals.