What if dog is obsessively licking their mouth and nose?

I'd take this one to the vet. My yellow lab used to do that and it wasn't until I noticed a lump under his chin area that I discovered he had lymphoma, a type of cancer. He was licking in response to a lymph node in his sinus area causing him discomfort.

I don't want to alarm you, but it's also possible that chemo and treatments can delay the offset, while extending their lives for quite awhile.

Then again, it could be something as simple as an allergy. You really need a vet to check the animal.