A sleeping cat is always super adorable. What takes that cuteness to derp level is the sight of that dozing cat with their tongue out.
Unless your cat has health issues, is missing front teeth that would normally restrained the tongue inside the mouth, or is feeling excessively hot, sleeping with his or her tongue out is a harmless thing to do.
Cats who sleep with their tongues sticking out are deeply relaxed
In the same way your mouth might fall open when you are napping, a cat’s tongue will slid out of a relaxed cat’s slightly open mouth during sleep. This happens when the cat’s jaw relaxes during deep sleep, causing the mouth to slightly open and the tongue to fall out.
On the internet, the act of a cat with their tongue sticking slightly out is called a blep (not to be confused with a mlem).

For some cats, sleeping with their tongue out helps them to breathe
Cats may stick their tongues out slightly to facilitate breathing, especially if they are in a particular sleep position that may obstruct normal breathing patterns. This behavior might be more common in brachycephalic breeds, which have shorter muzzles.
Not all cats sleep with their tongue sticking out
Not all cats will sleep with their tongues sticking out. If you have more than one cat as a pet, you might notice that only one of them will sleep with his or her tongue out and the other one never does.
Each cat has his or her own quirks and sleeping with a tongue out is one of them.

Your cat may even continue to sit there with their tongue sticking out after they have woken up from their snooze. Some cat experts believe the quirk of a cat with its tongue sticking slightly out of his or her mouth is a sign that your cat feels very comfortable and safe.
When to be concerned about your cat sticking their tongue out
If your cat is sleeping peacefully and his or her mouth is still and relaxed, there is nothing to worry about. If your cat seems agitated and is sticking their tongue out and has with an open mouth or is panting, that could indicate distress.
Oral discomfort or dental diseases might cause a cat to sleep with its tongue protruding. If this behavior is coupled with other signs such as drooling, reluctance to eat, or pawing at the mouth, you should consult with your vet about your concerns.
Likewise, if your cat is constantly moving their tongue in and out of their mouth, that could be a sign that there is an object inside the cat’s mouth that they are having trouble removing or that there might be something painful inside the mouth.

If your cat’s face or paws are twitching, but your cat’s mouth remains relaxed, they might just be dreaming.
As with all health concerns, if you aren’t sure what is going on, you should consult with a veterinarian.
Take a blep picture of your cat
If your cat does sleep with their tongue out, take the opportunity to take a cute picture. Reddit even has a whole section to enthusiastic cat owners sharing the best help posts of their sleeping and awake cats.
References
Gleason, H. E., Phillips, H., & McCoy, A. M. (2023). Influence of feline brachycephaly on respiratory, gastrointestinal, sleep, and activity abnormalities. Veterinary Surgery, 52(3), 435-445. https://doi.org/10.1111/vsu.13931
This article was originally written on July 21, 2021 and has since been updated.