The Febreze web page that addresses this rumor notes:
Since December, 1998, there have been Internet rumors circulating stating that Febreze may not be safe to use around pets. These rumors are not true. Used as directed, Febreze is safe to use around pets.
Febreze Fabric Spray has been safely used in 40 million pet owning homes around the world.
Febreze was tested for nearly 5 years by scientists, doctors, safety experts and veterinarians, and all have come to the same conclusion: Febreze is safe to use around pets.
The American Society for the Prevention of Cruelty to Animals (ASPCA), the nation's leading authority on pet safety, has investigated these rumors and issued the following statement:
"Based on a thorough review by veterinary toxicologists at The ASPCA National Animal Poison Control Center, together with other outside experts, The ASPCA considers Febreze safe in households with dogs and cats when used as directed."
This information is aimed primarily at dog and cat owners; Febreze acknowledges that consumers should be cautious about using Febreze (or any other aerosol or spray product) around birds:
However, as birds are uniquely sensitive to some airborne household products and environmental factors, experts recommend removing the bird from the room until the product application has fully ventilated, like you would do with any household cleaning product.
Of course, the statement that Febreze is "safe" assumes that it is being used with normal due care by pet owners; nearly any product can be dangerous when misused. (Lemon furniture polish is perfectly safe if it's used to polish furniture, but not if it's used to make lemonade.) In this case, "due care" means not spraying it directly on pets, and removing birds from the room during its application.