What possible scientific reasons could there be for a vampire to only be killable via wooden stake to the heart?
This excludes metal stakes, knives etc. to the heart, it's not the bleeding out that's the issue. Just for some reason, a wooden stake to the heart ends a vampire. Why?
Assume the type of vampire that is common in fiction (can't go out during the day, doesn't like garlic or crosses, but can pass as human), no Twilight sparkle people please. Vampirism in the context of this post is spread via a transmittable virus that lives in saliva and blood.
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Any scientific reason would have to be in the make up of wood itself: See The Chemistry of Wood.
Perhaps because wood was living cells; something the cells produce; everything from sugars (like in sap) to cell walls -- or a "recipe" of such products in a particular balance. Think of a chemical recipe as producing a kind of molecular machine, and a machine that won't work if any element is too much or too little: Just like a cake won't rise correctly if it has too much or too little liquid, flour, baking powder, salt, sugar, etc.
Or the science can be in the actual DNA of the wood: DNA and genes are physical things, and the DNA unique to woody plants could be the machine that is causing the vampire death.
In this case for vampire lethality, "wood" must have a precise combination of cellulose, lignins, and other atoms in precise position and combination to cause an interaction with the vampire heart that degrades it and poisons the rest of the vampire system.
If you want the vampire to be dusted, "Scientifically" we'd say whatever biological compound was holding the vampire cells together was defeated by the wood, and the unraveling of each cell causes neighboring cells to unravel: thus a massive chain reaction that unravels every cell of the vampire, effectively turning it to dust. Such chain reactions are not unusual at all in chemistry.
Why the Heart? Well, the heart is obviously specialized cells that do not occur anywhere else in the body; so it has something to do with the nature of those cells. Perhaps they are uniquely vulnerable to the effect of the wood, the only cells in the body with the particular atomic configuration that the wood "fits" on the outside of the cell; so just touching wood gets the wood "recipe" inside the heart cell; where the cell replicates it (like a virus) to the point of rupture: then the physical force of that rupture infects other cells, which rupture, and voila! exponential growth of a disintegrating core; a chain reaction.
Of course if there is science, there is a chance of isolating the compound in wood (by experimenting on vampires, or vampire heart tissue), so something like steel stakes or bullets could be coated in it; which might make for easier delivery. But it still must come into direct contact with a (formerly) beating heart cell, so must penetrate the chest. But something like bullets might be needed if a vampire had (through surgery) encased his heart in a hardened steel shell.
Vampire hearts typically do not beat; so they don't circulate blood. So a wooden stake elsewhere in the vampire body touches no heart cell, does not circulate to the heart, and therefore the special ingredient in wood causes no disruption. In fact other cells could metabolize that wood compound (break it into pieces) and render it harmless.
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I agree with @Separatrix that the wooden part may be optional.
From a pseudo-scientific point of view, staking the heart may be used to stop the vampire from regenerating. As cutting the head off lets you eliminate the brain of the vampire and kills of the body, stopping the heart may stop the vampire itself.
Now, undead vampires are often described as having no pulse, but that makes sense only in a more "magically explained" context. From a more scientific point of view, all vampires still need blood to survive. It is only natural to suppose that this blood is carried around in veins and arteries.
Since blood is the source of power, stopping the heart would mean stopping blood flow - so movement, healing capabilities, strange vampiric powers ... the vampire itself may not die and fall into a dormant state, if you wish, but if it can't move it's almost as good as killing it (bonus points: it's easier to cut off its head). The important thing is that the vampire should not be able to remove the stake by himself.
Bonus points, maybe wood also poisons the blood, making the heart stop faster or impairing movement better on elder/more dangerous vampires.
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