If you just want the concept, there are pasta sauces involving egg or egg yolk where the heat from the pasta is the only thing that cooks the egg.
In that way you cooked your food with food, but you didn't have to do without either piece of it.
cooking.stackexchange.com/users/304/Kate Gregory
It's possible to smoke meat using nuts or herbs, and fish can be baked on a huge slab of rock salt, but other than that, people don't really bother.
There's nothing wrong with experimenting though.
Maybe you'll invent something new by burning fruit.
Who knows.
cooking.stackexchange.com/users/54916/neko
In a way, exactly that happens in a flambe - potable alcohol (you can drink it and it has calories - food ;) ) is set on fire to cook stuff... and in some preparations (eg very high heat wok cooking), cooking oil (edible calories too) are set on fire for a moment...
Of course, there is a symbolic/style/fashion/hip/just plain decadent factor to take into account here - a burger prepared that way would make people CLAIM it tasted better, and that make enough people PERCEIVE it as tasting better, no matter what the actual influence of the technique is - unless it renders the food patently unpalatable, unsafe (though that has a hip factor in itself) or inedible.
Also, in a way, any smoking wood you use IS a food ingredient:
You process it (by burning it) to extract an edible/palatable fraction (the smoke particles) and add that fraction to your food (by exposing it to the smoke).
cooking.stackexchange.com/users/35312/rackandboneman
What about acidity to "cook" fish ?
You can make ceviche by "cooking" the fish in lime and/or lemon juice. or salt to cure fish and meat ?
You can make gravlax by "cooking" the fish with sugar and salt.
cooking.stackexchange.com/users/20118/Max
Stakes may burn well, steak does not.
Go ahead, light a pile of them.
I dare you.
How long did those burn for before they fizzled out?
I generally try to avoid having fat-flare-ups when grilling, as the thick greasy black smoke is the opposite of an improvement in flavor.
That would be most of what you'd get from tossing a steak on the coals, which is about the only way you're going to "burn steaks" without having them either not light at all or go out in seconds, depending how hard you try to light them.
Your fuel would still be charcoal.
The only "practical" application of your general concept which comes to mind (only "practical" due to weird US farming subsidies) would be using a corn-burning stove, and I doubt that would have any particular flavor benefit.
You could burn other dry grains, pasta, or nuts if money is no object (presumably if you are starting with "burn steaks.")
The nuts will tend to greasy black smoke again (I once gave up on shelling some black walnuts and used the remaining ones for firestarters; They burned quite well.)
cooking.stackexchange.com/users/34242/Ecnerwal
There is a self -fueling food, but it's only partially self-fueling.
You need another heat source to provide the ignition.
First, get the fattiest grind of ground/minced beef that you can find.
Cook it over either charcoal or a gas grill with a heat deflector.
Place the burger directly over the charcoal or heat deflector.
Close the lid, which will help heat the entire burger so it renders the fat.
As the fat drips, it will create flare-ups which will cook the burger faster than just the starting heat source.
It helps to use thinner patties -- as all of the fat is then relatively close to the surface, a higher percentage will drip out ... and the thing will be cooked before the outside becomes a hardened husk.
cooking.stackexchange.com/users/67/Joe
I was thinking about it and we use charcoal, gas, wood chips, etc to heat our food; for obvious reasons and since they burn for a long time.  But what if, for example, you set a few steaks on fire and use those steaks to cook a hamburger? Would it improve the taste of the hamburger?
Using food as a cooking fuel to cook other food?
cooking.stackexchange.com
<flavor><food-science>
cooking.stackexchange.com/questions/78721
cooking.stackexchange.com/users/54914/Katianie
Steaks would not burn well, but there are a number of examples of food cooking in other food. Eggs or egg yolks can be cooked in pasta, meat can be smoked with nuts or herbs or on rock salt, flambeed food is cooked with cooking oil that burns briefly and ceviche can be made by "cooking"  the fish in lime and/or lemon juice. There are many examples of a food used to cook anoth food, pasta carbonara for example. However, wood stakes burn beter than meat steaks so it is probably better to stick with them.
There are numerous examples of one food being used to cook another food. These include pasta carbonara and ceviche. Wood stakes burn well, meat steaks do not.