The color of the dye is usually visible because it has some 'tail' of absorption spectrum in visible region. But IR absorption is much stronger, it is not visible. Upon excitation, absorption spectrum changes but it cannot be seen because 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 fluorescence of other fluorescent dyes, that absorb and emit visible light.