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Is there anywhere that service dogs are not permitted?

Generally guide, hearing and service dogs are permitted to accompany their disabled owner everywhere members of the public are allowed, but there are a few exceptions. For example, a member of the public would be permitted in the dining area of a restaurant, but not in the kitchen. Therefore, a guide dog would be permitted to accompany his disabled owner in the dining area of a restaurant, but not into the kitchen where food is prepared and special clothing and sanitation procedures are required.
It is also an important distinction to note that it is the handler who has access rights and not the dog. A guide dog without his blind handler has no particular access rights of his own and neither does a hearing dog or other service dog without his disabled handler.

"Under the ADA, State and local governments, businesses, and nonprofit organizations that serve the public generally must allow service animals to accompany people with disabilities in all areas of the facility where the public is normally allowed to go. For example, in a hospital it would be inappropriate to exclude a service animal from areas such as patient rooms, clinics, cafeterias, or examination rooms. However, it may be appropriate to exclude a service animal from operating rooms or burn units where the animal’s presence may compromise a sterile environment." -- U.S Department of Justice ( http://www.ada.gov/service_animals_2010.htm ).

For clarification, contact the U.S. Department of Justice's ADA Information Line at 800 - 514 - 0301 (voice) or 800 - 514 - 0383 (TTY)
"When it is not obvious what service an animal provides, only limited inquiries are allowed. Staff may ask two questions: (1) is the dog a service animal required because of a disability, and (2) what work or task has the dog been trained to perform." (see 2010 guidance linked above). If these questions are not appropriately answered, the business may exclude the animal, but not the person.

Though service animals of all kinds can legally accompany their disabled handler almost anywhere the handler goes, they can be excluded from areas where their presence would constitute a fundamental alteration of goods and services available for all customers, an undue burden, or a direct threat to safety. Some exceptions to ADA access rights are due to other federal laws, treaties, or the Constitution. For example, the exclusion for churches comes from the First Amendment to the Constitution. Other exceptions are written into the implementing regulations for the ADA itself in sections 36.208 (direct threat), 36.302 (fundamental alteration), and 36.303 (undue burden). Examples where a service animal might be excluded include (but are not limited to):

-Sterile rooms, such as operating rooms, some areas of emergency rooms/departments, some ICU rooms, some ambulances, some delivery rooms (on a case-by-case basis based on actual risk assessment)
-Clean rooms where microchips are manufactured
-Places where food is prepared (though they cannot generally be excluded from dining areas where food is present) (by order of most health departments)
-Open air zoological exhibits, such as open air aviaries or butterfly gardens (at the zoo's discretion but based on actual risk assessment)
-Churches (at the church's discretion)
-Native American Tribal Council Chambers (at the council's discretion)
-Federal Courts (at the judge's discretion)
-Jail or prison cells (at the discretion of the facility director)
-Private clubs (at the club's discretion)
-Private homes (at the home owner's discretion)
-Some amusement park rides (at the park management's discretion but based on actual risk assessment)

If a business can show that allowing a service dog to enter a specific part of their business would constitute a direct threat, fundamental alteration, or undue burden, then they may legally exclude service dogs from entering there, but must still accommodate the person with a disability without their service dog.

For an example of a direct threat, consider a burn unit or ICU caring for a patient in very fragile condition where doctors tell us that the mere presence of a dog, even a clean, well-behaved dog, poses an unacceptable risk to the life and health of the patient because the risk of exposure to loose hair or zoonosis is in their medical opinion too high, then a service dog might be excluded from the burn unit or ICU. A general rule of thumb (which still has exceptions) is that if special clothing such as gown or gloves is required, then a service animal might also be excluded and certainly a service dog handler upon noting special precautions are required for a given area should inquire about whether a service dog can enter without endangering the patients within.

For an example of a fundamental alteration, consider a cat rescue society that attempts to find homes for abandoned/neglected/abused cats. They might reasonably exclude service dogs from their adoption area because the mere presence of a dog is known to cause distress to some of the cats up for adoption. They might instead offer a secure location for the service animal while the human partner visits with the cats or remove individual cats for inspection in another area, away from those that are afraid of dogs. The ultimate purpose of such a facility is to rescue cats from trauma and traumatizing a cat by exposing it to something it fears would fundamentally alter that purpose.

For an example of an undue burden, consider a swimming pool which uses special filtering equipment and chemicals to maintain a certain level of cleanliness and hygiene for swimmers. If they know that their filters cannot handle the added load of dog hair or that adding a dog to the pool would upset the delicate balance of pH and chlorine, thereby putting swimmers at risk of disease and that it would cost them more money than they can afford to upgrade the pool facilities to also handle canine users, then they might legally exclude a service animal from entering the water, but not from being in areas of the facility where people are permitted in street clothes and shoes, such as possibly the decking around the pool.

So far, this discussion is centered entirely on laws of access in the United States of America. Other countries will have their own laws in place regarding the access rights of individuals accompanied by a service animal.