There are concerns regarding the use of psychiatric drugs including benzodiazepines in pets and animals. According to Merck Veterinary Manuals, there is little evidence to support psychoactive drugs and sedatives in veterinary medicine:

Anxiolytics, antipsychotics, antidepressants, and mood stabilizers used to treat human behavioral disorders are being used more commonly in veterinary medicine as adjuncts to behavioral modification therapy (also see Principles of Pharmacologic and Natural Treatment for Behavioral Problems). Few veterinary clinical studies have been reported, and guidelines for veterinary use are grounded on therapeutic applications in human medicine. – Merck Veterinary Manual

The risks of toxicity, Serotonin Syndrome, and some other adverse effects are mentioned.

Since our pets can’t tell us what sort of adverse effects they’re experiencing from medication, they may go unnoticed. However, it’s up to people to recognize them. Sometimes it can be difficult to detect adverse effects of a medicine in some animals .

If long-term use of sedatives and other psychoactive substances in humans can potentially result in tolerance, dependence, adverse effects and a withdrawal syndrome necessitating slow tapering in humans, then the Merck Veterinary Manual implies that this risk can be extended to animals. Additional issues that may arise for animals on these types of medications may also be possible, e.g., certain types of small pets and horses have a unique gastrointestinal anatomy that may be adversely affected by sedatives in ways that may not be considered in humans.

For these reasons, use of long-term BZDs in animals is not supported by W-BAD, and similar medically necessary medications should be managed carefully by an experienced veterinarian.

Consult your trusted veterinarian before starting, stopping or making changes to your pet’s medication. Keep medication out of the reach of children and pets.

Further reading:

Merck Veterinary Manuals: Tranquilizers, Antidepressants, Sleep Aids, and Anticonvulsants (Toxicity)
Veterinary Medicine: Poisoning of dogs and cats by drugs intended for human use
New York Times: Pill-Popping Pets
anh-usa.org: Prozac for Pets (includes petition)