Well.....OK see below, but I don't EVER remember Kirk firing Phasers at warp in TOS. Help me out here.
Anyway from Ex Astris Scientia (
http://www.ex-astris-scientia.org/inconsistencies5c.htm)
Phasers
Phaser beams There is only one definite fact about phasers, that they release stored energy and need to be recharged at latest after some shots, as opposed to lasers whose energy output is continous. A phaser beam probably doesn't consist of charged particles (ions) either, because the beam would quickly disperse due to the mutual repulsion of equally charged ions and because it would be rather easy to shield a ship against them with a static electromagnetic field. The shields in Star Trek, however, are always high-frequency fields, and they are probably not electromagnetic, but of gravitational nature. It is even more nebulous of what exactly the beams of phasers consist since VOY: "Scorpion" when the weapons were suddenly capable of firing macroscopic objects in the form of nanoprobes. No matter how hard I try, I can't imagine how they could survive in there, unless the phaser worked similar to a transporter, meaning that some sort of annular confinement beam can be added or is intrinsically available, just as the transporter beam that keeps macroscopic objects in one piece.
Phasers at warp
We have occasionally seen phasers being fired at warp, like in VOY: "Message in a Bottle" or DS9: "Treachery, Faith and the Great River". If the phaser were anything like a laser in that it released photons, it would be restricted to exactly the speed of light. A particle beam consisting of ions, on the other hand, could reach any desired speed below c. A confinement beam, like already mentioned above, could be the solution to this problem too. If this beam extends the warp bubble around the phaser beam, the beam itself would be able to travel at FTL speed. A similar technology is used for FTL processing in 24th century computer cores, according to the TNGTM, and the warp sustainer engine of a photon torpedo (see further below) may work like this too in that it "grabs" the warp field from a starship. The problem is that this kind of confinement would have to be still another one than the confinement beam that would allow objects like nanoprobes to "ride" on a phaser beam.
Probably in response to previous inconsistencies, the DS9TM explicitly mentions the possibility of firing phasers at warp, thanks to the recent development of an "ACB-jacketed beam device".
BRING ON THE ARGUMENT!
--SghnDubh