Cancer: Citrus Red 2, Red 3, Red 40, Yellow 5, Yellow 6, Blue 1, Blue 2 and Green 3, — which include some of the most commonly used synthetic food dyes — have been identified as being, or being contaminated with, potential cancer-causing chemicals. The potential carcinogenity is largely based on small-scale animal testing in laboratories.
Allergies: Blue 1, Red 40, Yellow 5, and Yellow 6 are known to trigger reactions in those with allergies.
Hyperactivity: In 2009, Britain asked food companies to phase out the use of most food dyes in children's food over concerns some studies had raised about their link to hyperactivity, attention deficit disorder and related health issues. (Other experts in the field have questioned the results of those studies.)
Of course that is eating food dyes. We don't know the effect of inhaling them.
Still, food dyes are GRAS - Generally Regarded as Safe.