In Star Trek TOS, whenever a human is struck by a phaser beam set to kill, they glow and disappear. They never leave any residue or indication they were ever there. When they are hit by a phaser set to stun, the people drop with no sign of being burned. What happens to people when they are hit by a phaser beam? Why aren't they burned on stun, and where do they go when they are killed?
Phasers have an entire spectrum of settings. They fire a fictional particle called a nadion. At its lowest setting it disrupts a life form's nervous system, either knocking it unconscious or cause it to become dazed. At its highest, it is enough to vaporize the targeted object. In between there are settings for killing, cutting, wide-dispersal beams etc.
The stun setting is capable of causing trauma/burns or even killing the target if fired at extremely close range, if fired at a vital organ or if exposed to a stun beam for too long.
The vaporization setting has rarely seen action in TNG and later series, which were somewhat more 'civilized' compared to the original series.
You seem to be looking for a Real Physics™ explanation for what phasers seem to do, but I doubt that there is one. If phasers "vaporized" a victim by heating them until their solid and liquid components became a gas, you'd get a great big explosion because high temperatures + low local pressure = large volume. But that's not what we see as a result of phaser fire.
If phasers got rid of their victim's mass by total mass conversion (E = mc²), you'd get a bigger and nastier explosion, consisting of hard radiation, energetic particles and ionized gas. How bad would this be? Imagine piling up a quarter of the world's total current nuclear arsenal and setting it off all at once. That's what phasering a 75kg man would be like.
If Star Trek phasers worked like Known Space Slaver disintegrators, i.e. depressing the electrical charge of the electron, you'd get plasma without the high temperatures that would produce gases, thus you'd end up with lots and lots of dust. But that's not what we see as a result of phaser fire, either.
The only theory I can think of that makes sense in Star Trek Physics is that phasers fired at full power dump their victims into subspace. The glowing corpse simply topples away into another dimension, with no pesky explosion or shower of hard radiation to muss our smooth-limbed heroes.
There was a "Writer's Technical Manual" given to the writers for Star Trek: The Next Generation, written by the show's technical consultants Rick Sternbach and Mike Okuda, which was later expanded into a published book called Star Trek The Next Generation: Technical Manual (also written by Sternbach and Okuda). The original version for the writers is a little bit easier to follow since they don't add as much semi-meaningless technobabble, so I'll quote from that, since a copy of the fourth season edition has been posted online here. In the section on phasers on p. 34 of this edition, one relevant thing they say is this:
We won't go into the exact nature of the energy beam; suffice it to say that the beam manages to excite water molecules like a microwave oven (at the low end of the scale), up to being able to negate the nuclear forces holding matter together (at the high end), and various steps in between.
So, we can assume that when people appear to completely vaporize, it's at that high end where the phaser is disrupting the nuclear forces that hold the nucleus of each atom together. In real life, the only process we know of that disrupt nuclei also happen to cause nuclear explosions, but this is some weird future technology so maybe it doesn't work the same way.
Later on the same page they list the effects of different phaser settings--they say that "nuclear disruption energies" first come into play at setting 6, and then at setting 8 they say:
Setting 8: TOTAL "ZAP"! This setting will cause a human or humanoid to totally vaporize, as physical forces break down. About fifty percent of the disrupted matter disappears from this continuum as high-energy converted particles (the reaction takes place in a very short distance from the target, so thermal effects travel only a few inches). The remainder of the target becomes water vapor, carbon dioxide, calcium phosphate, and various other compounds.
I'm not sure, but "disappears from this continuum" might suggest half the particles fly off into subspace (with the remaining particles no longer being bound together by molecular bonds and just dissipating into their surroundings, I guess), since subspace has at times been represented as some sort of other dimension not part of our normal space (or, possibly, some sort of energy field that exists in such a dimension). For example, in the TNG episode "Schisms", LaForge and Data find a "subspace particle stream" made up of "tetryon particles" which are "emanating from a tertiary subspace manifold", and Picard says "I thought that tetryons were unstable in normal space" and LaForge replies "We don't understand it either, sir. Something from that deep in subspace shouldn't be able to exist in our universe". This seems to indicate that the "subspace manifold" is not part of "normal space" or "our universe", and indeed, it's later revealed in the episode (I guess this is a SPOILER if you haven't seen it, but the episode has been out so long I don't think I need to put this in hidden spoiler text) that the particles are actually being emitted by some weird alien beings living in that "subspace domain", so again it sounds like a sort of parallel universe in another dimension, perhaps a bit like two parallel 2D planes at different locations in a higher 3D space (the scene directions in the script say of the aliens' domain, 'We are in another space -- tetryon space'). Similarly, in the episode "Force of Nature" Data says that damage to subspace caused by warp engines could cause a disaster in which "subspace will extrude into normal space, forming a rift", again indicating that subspace is usually separate from "normal space". Finally, the TNG series bible, written by Gene Roddenberry as a guide to writers, refers on p. 40 to "subspace radio which operates through another dimension of space", showing that this idea of subspace as another dimension was indeed Roddenberry's intent.
Also, for comparison, p. 135 of the published version of the technical manual says this about the effects of high and low phaser settings:
As with the ship's main phasers, the greater the energy pumped into the prefire chamber, the higher will be the percentage of nuclear disruption force (NDF) created. At low to moderate settings, the nuclear disruption threshold will not be crossed, limiting the phaser discharge to stun and thermal impact resulting from simple electromagnetic (SEM) effects.
There's no real discussion of why the "simple electromagnetic effects" result in humanoids being knocked unconscious, they just say that setting 1 is "calibrated for base humanoid physiology, and causes temporary central nervous system (CNS) impairment." (The TNG series bible I mentioned above does say on p. 43 that the phaser beam "scrambles the flow of electrical currents and light fiber currents", so since nerves in our body communicate with electric impulses, this might explain the stunning effect.) There is no mention of a single shot of a phaser blast causing burns until setting 4, at which "base humanoids experience extensive CNS damage and epidermal EM trauma", though it does mention that "long exposure" to setting 2 can cause "epithelial damage" (i.e. damage to skin). As for high settings, p. 136 of the published manual says something similar to the writer's manual about particles disappearing from "the continuum" at high settings:
Setting 8: Disruption Effects; discharge energy 15,000 for 1.75 seconds, SEM:NDF ratio 1:3. Cascading disruption forces cause humanoid organisms to vaporize, as 50% of affected matter transitions out of the continuum. The damage index is 120; all unprotected matter is affected and penetrated according to depth/time.
They vaporize. The magic behind the stun setting I've never managed to figure out. It's effective on practically every humanoid but the tougher ones get stunned only at above normal settings. It's actually never verified whether or not people receive stun-burns although I've heard it being said in disgust that someone was "stunned multiple times".
Seeing by the controls it's a lowest power setting, perhaps they overload nerves.
Note that not all ST weapons give so clean a vaporization as Federation issued phasers do, disruptors especially leave quite the typical mess.