Adding hair color can dry out your hair, but it does not make it fall out.
Permanent or oxidation hair color techniques are the only types that give long-lasting coverage for gray hair found on more than a third of the scalp.
After coloring, the hair can be drier and more brittle since the scales that cover the hair shaft are opened out during the coloring process to allow pigments to penetrate the hair. The roots are not affected however. It is recommended that you use hydrating and detangling products.
It is possible to develop an allergy to hair coloring products, which is why your hairstylist usually tests a section of hair behind the ears first. In the case of an allergic reaction, you can get tested by a dermo-allergist.