Sometimes no and sometimes yes
Probably the closest equivalent in spirit to the dye arrow is the net, a weapon that, on a successful ranged touch attack, has an effect yet deals no damage. A GM that's trying to determine if an effect should work in conjunction with a dye arrow should ask if he would allow the effect to also work in conjunction with a net.
Like the net, a dye arrow deals no damage. In fact, the dye arrow's described as not possessing a pointy bit at the end, and, like most arrows, a typical dye arrow is instantaneously destroyed upon making with it a successful attack. With that in mind, this GM would likely forbid any effect that he believed relied on the arrow dealing damage to generate that effect with a dye arrow. So, for example, in this GM's campaigns, a rogue couldn't deal her sneak attack damage with a dye arrow any more than a rogue could deal her sneak attack damage with a net.
However, this GM would allow magic weapon special abilities (e.g. flaming, shock) and other effects that activate merely on a successful hit to activate when used in conjunction with a dye arrow.
That is, in the same way that this GM would allow a successful attack with a +1 flaming net to deal its +1d6 points of extra fire damage to the gladiator's foe, an archer launching at her foes +1 flaming dye arrows could deal the extra +1d6 points of fire damage with that ammunition. (The typical PC archer will likely find regular arrows deal more damage, but if equipping peasants to combat El Guapo or whatever, +1 flaming dye arrows may be a thing.)
Note: In case the GM doubts the legality of the dye arrow, the Pathfinder version of the dye arrow—it having originally appeared in Paizo's OGL product Elves of Golarion whence d20PFSRD takes its dye arrow description—appears in the Alchemy Manual (21). Cost and description is largely unchanged from its original appearance.
