I know that there are several saliva glands in the mouth, many or most of which are active at any given moment – this question isn’t about that. Does the skin inside a person’s mouth have any active sweat glands? Does the phrase “lip sweatin’ hot” carry any physiological legitimacy?
The Dun King
The oral labia, or mucous membrane, don’t sweat. They produce mucous. The phrase you cite actually refers to sweat forming on the region between the upper lip and the nose, where there are sweat glands. IIRC.
QtM, MD
Wow, thanks for the quick reply! 
I’m going to expose my ignorance of anatomy here…
I had thought a mucous membrane was a globby clump of mucous – like membrane of snot – but it looks like it’s a gland or many glands lining the skin and other tissues that secrete mucous. That’s as far as I’ve gotten in 10 minutes with Google.
Is the mucous membrane one gland for the whole body, or are there many? Is there a mucous network that spans the body like the lymph system?
Is the mucous membrane always produsing mucous for the mouth, or does it fire up for special occasions? And is this mucous the same stuff that I see when I cough up a loogie?
The science of snot, it’s an awesome thing. 
The Dun King