How do I stop the hair-raising, tongue-numbing burn of spicy food?

When my husband and I visited a local Schezuan restaurant he got a hold of one of those nasty ghost peppers, popped in his mouth and before he realized what he had done, there was a 5 alarm fire going on in his mouth. It scared me because he couldn't talk and was in some distress. The wonderful waiter saw what was happening and quickly came over with a bowl of sugar packets. He said take these until it stops. After 3 o4 packets he was fine. The trick is hold the sugar in your mouth till it dissolves. I also learned this trick when making Habanero jelly. It's true dairy helps too but when it's not available try the sugar thing. I always keep a couple packets in my purse now... you just never know.

I use similar techniques to those recommended by Lori T and Miss Karen, so nothing to add there.
But I would suggest a cautionary note and technique to prevent the problem arising.
When making a recipe new to you with hot and spicy elements, at first only add half the specified amount. Taste as you go, adding the remaining hot spice later in cooking or when serving, for people yo add to individual tastes.
This (defensive) approach will prevent overseasoning that may occur because of differences in taste between you and recipe writer, your spices or hot sauce being fresher or stronger, even typos and proofreading errors.
After you've made and liked a recipe once, note the amount of chilies or hot sauce that you used for repeated cooking.

I don't know about 'saving'' or 'fixing' the mouth on fire dishes, but the extinguisher(s) are:
Milk products, bread or honey. The flame throwers are water or alcohol.... BTDT.

I made that OMG it's too hot mistake once, when I made Jerk Chicken for the wonderful man I eventually married. It was so hot neither of us had more than a bite, and I was afraid to give it away. So far as I know, there isn't any way to actually remove the extra heat from a dish. You can add dairy products sometimes, dilute a sauce with a second batch of "no spice added", try adding sugar, or extra starches like potatoes. If you have set your mouth on fire due to capsaicin, the best remedy I know of is a mouthful of yogurt to coat all the surfaces of your mouth, followed with a big bite of bread, and washed down with a glass of milk. That will bind to the oils burning your mouth, and let you get them under control. Anytime you deal with peppers, it's best to taste a tiny sliver or bit and determine how hot it actually is. Then it's best to err on the side of caution, because you can add more heat at the table, but you can't get it out. You can add stuff to help cover it, but that will always change the balance of taste in the dish. Otherwise, if you end up with a dish that's too hot for you, you have to run through your list of friends and hope there is a dedicated chile-head on it who will appreciate the gift of a ready cooked meal. If not, then you are kind of stuck because the only animals that can tolerate capsaicin are birds. Well, and a certain variety of Chinese shrews. We humans and those odd shrews are the only animals in the world who deliberately seek out hot peppers for an enjoyable burn.