When I drink my hot tea, and when I eat anything, I get food, beverages, and saliva on my lips. That is what some people with cheilitis do, and that is what causes their lip skin to change color (darken) and stain when they eat. Eating substances that touch my lips creates a film of food, beverages, and saliva on my lips which is certainly very detrimental in my case to lip healing. The saliva contains powerful digestive enzymes which also attack the lip cells, because that's what it is meant to do. In my case, the leftover saliva and food particles cause my lips to feel irritated. Wiping my lips with a napkin (especially a dry napkin) after I eat and drink might get crumbs off, but it's not enough to get the dried substances off my lips.
Also, toothpaste is harsh and is known to be bad for the lip skin, and it actually causes some people's cheilitis all on its own, but I always get some on my lips when I brush my teeth, so I need to get it off after toothbrushing, and merely wiping it off with a towel does not do that - I can still see faint traces of the toothpaste when I do that.
It is a much healthier state of affairs to have a clean lip surface, and to reapply whatever medication I am using every time after I clean my lips, rather than go around all day without cleaning my lips off, and let all the other things my lips touch build up and irritate them.
The argument that something applied in the morning will stay there all day long, at a cellular level, even with the chemical and mechanical disturbance of eating, drinking, kissing friends on the cheek - whatever the lips go through in a typical day - contradicts all your prescription directions (from your doctors) about applying your various lip medicines 2 or 3 or 4 times a day. (I'm thinking tacrolimus, bariederm, hydrocortisone, whatever your doctors are giving you.) If a medicine stays on your lips for 16 hours, why would the doctors want you to reapply it during the day? You might ask your doctor if you put some acidophilus milk solution on your lips in the morning, whether it would stay there and be actively working all day long, given what the lips go through in a typical day.
The argument that washing wounded skin is detrimental to its healing holds to a point, especially if the wound can be covered and kept dry all day long, but when an injured area is contantly exposed to irritants, dirt, oils, and unsterile and corrosive liquids (which the lips are), I believe that it is healing and protective of that area to occasionally wash the lips off during the day. A dermatologist said on a doctor's discussion forum on the internet that he recommends this for his cheilitis patients and has seen positive results, and it made sense to me, so that is why I promote the idea.
The proof is in the pudding, obviously, and I have had better results with my lips when I do it, versus when I don't do it. They feel calmer, less irritated, smoother when I wash them 4-5 times a day. I went for a day last week without washing them, to see what happened, and by the end of the day they were drier, more scabby, more irritated, and it felt like it was the wrong direction for me to go in.
I don't mind what anyone else does - I just try to give out helpful information. You've got to do what works for you, no one else, and experiment with the various options before condemning them, so that you are making an informed decision about your health.