Food coloring usually consists of water-soluble
pigment molecules that disperse into anything with
water. Plants, like most organisms consist of a
large amount of water and so these pigment
molecules spread through the water in a plant and
can show you where water travels in a plant.
Physiologically (photosynthesis, metabolism etc),
it's unlikely that food coloring inhibits plants
as most food coloring is harmless.
*Bonus: Blue food coloring, for example,
is blue because it absorbs all other colors of the
rainbow and reflects blue light. That's why it
looks blue; that's the color of light that is
reflected into our eyes by the pigment molecule.