Is there a gameplay advantage to any of the dyes?

Solution 1:

Vanishing dye + PvP? Maybe?

Otherwise, no. Dyes change the color of your armor but that is the limits of their power. For the aesthetically inclined it helps them stay color coordinated but they provide no practical benefit other than being one further outlet for achievements.

Achievements related to dyes:

• Born to Dye - Dye an item.
• Color Coordinated - Equip your head, shoulders, torso, legs, feet, and hands slots with items dyed the same color.
• Made It Work - Equip your head, shoulders, torso, legs, feet, and hands slots with items dyed in all different colors.
• Rainbow Connection - Use one of all(?) the different dyes in the game.
• Pink'd - Defeat all the bosses with your equipment dyed entirely with Lovely Dye.

Solution 2:

Nope. Purely decorative, aside from a few achievements for using specific dyes.