Dyes can cause sneezing, coughing and watery eyes in people who are intolerant to it. The most common dyes are used in food, hair care and skin care products and even medicines. There are children who can have anaphylactic reactions to dyes that are in the FDA's approved list, so the intolerance is potentially life threatning. Epinephrine is prescribed in these instances.
There are hundreds of dyes that are approved by FDA. There are some for foods, others for cosmetics, drugs and medical devices. For a complete list refer to
FDA approved Dyes
Read the label carefully.
Some labels may just say artificial color and flavors instead of specific dyes. There is more legislation that is going to be in place that will help identify dyes more readily then is currently possible.