A phaser fires a nadion particle beam at its target. That sounds like inertial mass is involved so it doesn't reach light speed. Semi-Out-Of-Universe we see how fast a phaser beam is: I'd estimate something around 80 kph. You can see it moving with the naked eye. But that might be just a way to tell a story; to establish causality to the audience: "Look! First the beam is here, now it's here, now something's exploding." So the audience knows why the enemy ship explodes: It was a phaserbeam, going from here to there. So maybe we are just shown a "symbolic slomo" to let us know what's happening.
Are there any in-universe insights on how fast a phaser beam travels? E.g. can I escape (escape not evade) one on full impulse?