The most likely cause of a cat allowing its tongue to hang out is dental disease, as stated by Vetinfo. The cat may be feeling pain and irritation that causes it to hold its tongue differently than normal.
Cats are prone to dental and jaw problems in the same way as humans. However, pets cannot tell their owners when they are feeling bad and need to get a check-up at the veterinarian's office. It is up to the owner to take note of symptoms, such as the tongue hanging out, drooling and not eating well.
Usually, if there is a dental problem cat owners will notice other symptoms to go with the visible tongue, as stated by CatChannel. Periodontal disease causes problems with eating and bad breath. Severe gingivitis can also cause drooling, poor eating and the cat to paw at its mouth. Any broken teeth or TMJ problems cause pain and drooling as well. Oral tumors are considered a common cause of drooling. A condition known as uremia, caused by kidney failure, can lead to uncomfortable ulcers on the lips, gums and tongue.
Nausea may also cause the cat to pant or leave its mouth open with tongue out. This can be from just catching a stomach bug or something more severe, such as liver problems.