I know people always say ‘milk’ but that is not uber effective.
I’m assuming you need something acidic and/or lipid soluble to neutralize the burning, what can be used for that?
Would gargling with vinegar be useful?
I’ve heard sugar helps, but I’ve never tried it and don’t understand the chemistry of why it would help.
Not sure. I like the burn. I also like alcohol and I wonder if a couple of shots would lessen the burn.
It is a solvent after all. And besides, why not?
you can’t “neutralize” capsaicin. the best you can do is apply something that can carry it away, and being capsaicin is as you said lipophilic, something fatty might help. Vinegar and alcohol seem like they’d just augment one burning sensation with another.
of course, even if you eat/drink something which lessens the burn in your mouth, you’re still going to pay for it the next day when the spicy food you ate decides to make its exit.
Honey. It coats your tongue, protecting your taste buds from the capsaicin. It also dulls the heat by overloading your senses with sweet. I’ve not tested it personally (I don’t intentionally eat spicy food), but I’ve seen others use it with positive results.
The molecule has several reactive functional groups. Turning it into something else might count as neutralizing. However, I do not know that it is reactive under any conditions that are friendly for the human mouth.
Does anyone know how it is metabolized in the body? If it is metabolized in the body? Many of us know very well that metabolism, if it occurs, can be incomplete.
If it’s bad enough that you really need to neutralize the burn in your mouth–a napkin. It’s a bit gross, and obviously only works in areas you can reach to scrub, but a paper napkin will get a lot of the stuff off fast. (The idea behind the sugar suggestion might be similar–scouring the affected surfaces with the granular sugar.) Following it up with milk or yogurt should help with the rest, but you probably need to swish it around in your mouth a bit before swallowing to do any good.
All that’s assuming you don’t mean taking something to neutralize it farther downstream. I think you’re out of luck there. Read the warning labels. 
wax is better.
what is being sensed likely just needs to time out.
the oily spice in the mouth yet to burn can be carried away with another oily/fatty substance.
There was a hot sauce specialty store in my general regional vicinity that would offer small sized tootsie rolls for relief from a sauce sample that was too hot. Anecdotal all the way, I’ve never tried it.
Yes, peanut butter or avocado can be helpful to negate the burn.
Moving to General Questions from MPSIMS.
only if they’re insanity peppers, Marge.
Look to the spicy cuisines. Spicy Mexican food is accompanied by lots of fatty cheeses, sour cream, etc, to dilute the capsaicin, and tequila to dissolve it with.
Oil or alcohol.
While capsaicin is most often referred to as “fat soluble”, what it more technically is is a nonpolar molecule. Nonpolar molecules are soluble in other nonpolar molecules, like oil, and not nearly so soluble in polar molecules, like water. Alcohol has a polar and nonpolar end, so it will act as a solvent for both polar and nonpolar molecules.
So alcohol will do it, the stronger the alcohol (and therefore the less polar water), the better. Of course, too strong and you can’t keep it in your mouth without replacing one burn with another. So…stronger than beer, less strong than Everclear. Think…oh, tequila. Or strong margaritas.
Or you can take a mouthful of olive or coconut oils and swish for a few minutes and spit it out. It’s called “oil pulling,” and it’s all the rage in oral health these days.
When my hands burn from slicing chilies the best cure is to slather them with oil, rub a bit and then wash all the oil off with lots of soap & water.
Pretty much what I thought in post #2.
I’ve used lime pulp/juice for burning hands. I wonder if it would work for the mouth?
Hey, you even invoked my name! 
Wiseass answer: That sentence makes no sense.
Real answer: You want to go with something handy that you’d find in a Mexican restaurant so you don’t embarrass yourself. Sour cream works well, but you’ll get embarrassed anyway if you grab a chipful off your date’s entree.
When neither a solvent nor a diluent was available, I have had pretty good results from eating a slice of white bread.