Why is green fur not a thing?
The technical answer is: Because the coloration of skin and hair is done by the two forms of melanin: Eumelanin, which is dark brown to black and Pheomelanin which is yellow to red. This enables colors from white (not pigmentation) to black (dense eumelanin pigmentation) and also colors in between by different ratios of the two pigments.

The evolutionary answer is that it obviously made no evolutionary advantage to have other colors. If you look closer at this, you will see that most habitats are not green or only for a part of the year. If you think about african plains, these are mostly brown, something like artic landscapes are only green for a very short timeframe in the summer, for the rest of the year these are white to brown. And even tropical rainforests are mostly dark to brown on the ground. They would clearly stick out of the environment with green colored hair.

For all these different habitats the animals living there have adapted their hair color. It is not uncommon for artic animals to change their hair color with the course of the year, dark to brown in the summer and white in the winter (artic foxes and snow hare) for example. Animals which live in deserts will have a paler color than those which live in dark colored mountain areas. 

See this article for more information: "The Adaptive Significance of Coloration in Mammals"

Also interesting in this context is this article which looks at the reason for zebra stripes: "The function of zebra stripes"