"Mammals can make only two kinds of pigment: melanin (black or brown pigment) and the reddish-yellow pigment that red-haired people have.
Frogs, birds, and others in the tetrapod, or four-legged, world can't make green pigment, either.
It turns out that all color variation that we see in tetrapod animals is the result of different combinations of patterns of deposition and refraction of the same two types of pigments: black and yellow-red. A chameleon's color changes because of rapid shape changes in refractory cells in its skin, not by rapid production or release of an actual pigment. Frogs are green because of the pattern of refraction of blue light by special cells in their skin, which blends with their yellow pigment.
The colors of bird feathers are also generated by light refraction but by a different mechanism. The surface of feathers has microscopic ridges that form ordered tracks, much like the surface of a CD. The spacing of the ridges and the size and orientation of the pigment granules they contain determine the feathers' brilliant greens and blues (see links in reference 2). Tiny air pockets in feathers can add to color variation in birds. Light refraction is also responsible for the color of the human iris, which can range into deep blue or green hues. Another example is the sky-blue nose of the mandrill. For an exploration of the ways animals make different colors.
As you can see, the question is really why mammals don't have the ability to generate a variety of colors in their fur."