The color of the dye is usually visible because it has some 'tail' of absorption spectrum in the visible region. But IR absorption is much stronger; it is not visible. Upon excitation, the absorption spectrum changes, but it cannot be seen because the excited state only lives for nanoseconds, and the dye returns to the ground state emitting fluorescence. Dyes that change their color after irradiation exist; they are called photochromic dyes (Unfortunately, we do not have any in our catalog).
NIR dyes emit infrared that is invisible. But, of course, you can see the fluorescence of other fluorescent dyes that absorb and emit visible light.