Although domestic cats are usually close to humans, they are only rarely infected with an Influenza virus. This indicates that infection and transmission of seasonal human (or even avian) influenza viruses to cats are generally self-limiting. But you should never underestimate the Influenza virus.
However, in the case of Influenza A(H5N1) viruses, which is a highly pathogenic strain, several species of Felidae, including domestic cats and tigers kept in zoo’s, became victims during outbreaks in domestic poultry. A probable horizontal transmission among tigers was demonstrated in a zoo in Thailand.
Subsequent experiments have shown that these species are highly susceptible to the recently circulating Influenza A(H5N1) viruses. An experimental infection with Influenza A(H1N1) virus demonstrated that cats would get a respiratory disease and cat-to-cat virus transmission was indicated. Another experiment showed that cats could get ill after being infected with the highly pathogenic avian influenza A(H7N7) virus. In 2016 it was reported that an outbreak among cats in a New York City animal shelter was caused by Influenza A(influenza H7N2), a rare subtype that has not been found previously in domestic felines.
It’s no wonder then that scientists are now taking a closer look at the interaction between humans, cats and the Influenza virus. The available evidence suggests that cats are more than collateral damage in avian flu's deadly global spread and may play a greater role in the epidemiology of the virus than previously thought.
Another problem may arise from the Influenza A(H3N8) virus. It first appeared in horses but then jumped to dogs. To determine if something similar could happen with cats, scientists experimentally infected 14 cats with the equine influenza A(H3N8) virus. All showed clinical signs, shed virus and transmitted the virus to a contact cohort.
 Thanawongnuwech et al: Probable tiger-to-tiger transmission of avian influenza H5N1 in Emerging Infectious Diseases - 2005
 Kuiken et al: Avian H5N1 influenza in cats in Science - 2004
 Van den Brand et al: Experimental pandemic (H1N1) 2009 virus infection of cats in Emerging Infectious Diseases - 2010
 Van Riel et al: Highly pathogenic avian influenza virus H7N7 isolated from a fatal human case causes respiratory disease in cats but does not spread systemically in American Journal of Pathology – 2010
 Newbury: UW Identifies Flu Strain Affecting NYC Shelter Cats as H7N2 Influenza. See here.
 Kuiken et al: Feline friend or potential foe? in Nature - 2006
 Su et al: Equine influenza A(H3N8) virus infection in cats in Emerging Infectious Diseases - 2014