Do I have to allow a disruptive assistance animal to stay in my restaurant?

While you do have to allow entry to an assistance animal in most cases, you don't have to allow an out-of-control animal to stay.

The bottom line is that an assistance animal must be under the control of its handler. The individual handling the animal must maintain control of it through voice, signal, or other effective controls. An individual may not allow the animal to wander away and must maintain control of the it, even if it is retrieving an item at a distance. In a restaurant, this generally means you don't have to put up with it if the dog jumps on people, barks excessively, or wanders around toward other tables to lick up crumbs from the floor. 
If the behavior of a service animal is legitimately bothering other guests, there are constructive ways to deal with it without running afoul of ADA protection laws. 1) Make sure the other guests aren't just being difficult. For example, fear of dogs or not liking how they smell is not a legitimate reason, under the ADA, to ask a disabled person to take their dog outside. 2) If the complaints are legitimate, respectfully tell the dog's handler and work together to come up with a mutually agreeable accommodation. For example, if the animal is sitting somewhere that makes it difficult for another guest to get out of their chair, you can ask the animal's handler if there is another spot the animal could sit that would be safer and still allow it to do its job. 
A person with a disability cannot be asked to remove his service animal from the premises unless: (1) the animal is out of control and the handler does not take effective action to control it or (2) the animal is not housebroken. When there is a legitimate reason to ask that a service animal be removed, staff must offer the person with the disability the opportunity to obtain goods or services without the animal’s presence.
How do I know it's actually a service animal?
You may never know, so get comfortable with that notion. The US Department of Justice guidance says it is permissible to ask the person who has the animal the following questions when it is not obvious what service an animal provides: (1) Is the service animal required because of a disability; and (2) what work or task has the it been trained to perform.
Staff cannot ask about the person’s disability, require medical documentation, require a special identification card or training documentation for the animal, or ask that the animal demonstrate its ability to perform the work or task. Animals whose sole function is to provide comfort or emotional support do not qualify as service animals under the ADA.