Sure, this absolutely can be done. In particular, you cannot destroy elements using conventional means like burning, so all the initial elements are still in the mass. The question is whether or not this is a good way to extract what you want. There may be better chemical techniques to do the extraction that are short of burning. The branch of chemistry which deals with the optimal conversion of elements and chemicals to other useful substances is chemical engineering, and figuring out which process is "best" (e.g. fastest, cheapest, highest yield) is crucial for industry.
There is a very funny story related to this idea on the Periodic Table of Videos. A scientist was trusted with the entire UK supply of plutonium (10mg, a very small amount) during World War II. He accidentally spilled it on to his lab bench. He then took a saw, sawed a large circle out of bench, burned that wood, and recovered nearly all the plutonium! So yes, this works. Here's the video, and this story begins around 15:30: Plutonium Video
EDIT: As to your update, there are lots of methods available, but it depends greatly on the starting conditions and materials. For example, burning is a great way to extract carbon: burning hydrocarbons or carbohydrates burns off the hydrogen or water (respectively) and leaves mostly pure carbon.
But extracting metals might take more work. Often, metals exist as ionic compounds. For example, silver nitrate, $\ce{AgNO3}$, contains siver $\ce{(Ag)}$ in the +1 state. While it might be possible to dissociate the silver with high enough temperature, it is far easier to extract metallic silver by reducing silver in the presence of a more reactive metal. "More reactive" simply means a metal that will more easily give up an electron to the less reactive sliver, changing the silver ion $\ce{Ag+}$ to metallic $\ce{Ag}$. Silver nitrate is highly soluble in water, and if you simply drop in a piece of metallic copper $\ce{(Cu)}$ metallic silver will slowly come out of solution and plate the copper with silver, while copper dissolves and creates copper nitrate. The equation is:
$\ce{2AgNO3 + Cu->2Ag + Cu(NO3)2}$
but the crucial part is the ionic equation:
$\ce{2Ag+ + Cu->2Ag + Cu^2+}$
which shows that the silver ion is reduced to silver, and the copper metal is oxidized to copper ion.
Again, Periodic Videos has the footage of this exact reaction: Silver Video
Now I picked this example precisely because of this footage, but it's not the best example because silver is so low in reactivity that it exists metallically in nature (i.e. you can find lumps of silver if you are lucky!) But many metals are so reactive that it takes a huge amount of energy to extract them. Examples are sodium or potassium, which are so reactive they explode in the presence of water (go search YouTube and have a field day!). To extract them as metals commercially usually requires electrolysis, which uses electricity to force electron transfer which would not otherwise occur spontaneously.
And on and on with other methods. There is no universal extraction method, at least not yet, and this is why we have chemcial engineers to figure out better and better procedures to use.