No, they are air powered and do not use any black powder to operate, so arrow rifles are not considered firearms.
Compressed air is a low-velocity powerplant when compared to black powder. The projectile from an arrow gun flies at sub-sonic velocities. The projectile from a firearm flies at super-sonic velocities. This is why the report from a firearm is so loud. Arrow rifles use compressed air to launch an arrow between 300 and 1000 FPS. The effective range of most arrow rifles is less than 50 yards.