Vertebrates do not in general have much biochemical option for creating pigments other than blacks, yellows and browns. They also can fudge greys to some extent. As Mike correctly says, they sometimes get their hair coloured by algae, as in his example of the polar bear, but that was an unfortunate example, being in a sense slightly pathological. In the Arctic that doesn't generally happen, because if it does, it is too cold to encourage algal growth before it is time to shed the hair. So polar bears are generally a nice, sterile white, except where they have splashed their fur with brown or yellow food residues.
Sloths however, do routinely have green fur, courtesy of algae growing in grooves in the coarse hairs. It appears to be of some importance as camouflage.
You might rightly object that mandrills, turkeys, and various reptiles have vivid, even metallic colours of many shades, even blues, greens, you name it. Not to mention birds' feathers. True, but most of that is physical coloration, analofous to the colours you might see in a rainbow or an oil patch on water, or on a CD seen at the right angle. Grind the feather from a peacock train very fine, and you get a brown powder. The blues and greens vanish along with the microscopic structure. Most of the colours of vividly spectacular fishes or the irises of blue or green eyes of humans generally vanish shortly after dying, because they are based on an analogous mechanism, only in these cases from specially shaped protein molecules that lose their arrangements as soon as they are no longer manitained. Sometimes they even vanish in ill health, as maintenance lapses.
Reds in the skin often are simply the flushing of blood-rich tissues.
Some creatures, especially tree-dwelling snakes and lizards, are a vivid green that does indeed stem from a pigment, but it is a really weird example. The "pigment" in this case is a breakdown product of the haem in blood cells, called biliverdin, the green bile pigment. These creatures accumulate so much biliverdin in their blood, instead of excreting it by the bile or urine, that their blood goes bright green and shows as such through the skin!
I really gaped when I heard that one the first time. Occasionally I think of it again and gape again. showing my red tongue and gums that are red by virtue of the blood flowing through them and would turn blue if I were suitably hanged or poisoned with cyanide or any other process that left the blood short of oxyhaemoglobin (neither of which I intend to verify by way of experiment any time in the near future. Sorry!)
You will note that there is not much scope for hair to be coloured by such means, but there is one not very spectacular exception among our local, lovely golden moles. They are not very pretty, most of them (except for the silvery Namaqua dune mole that is stunning, I think), though they are very, very touching little things. They certainly are not golden. However, they have a slight iridescence to their fur that shows up as greenish metallic tints in the sunshine. That, like the peacock green, is a "physical" colour, not a pigment.
