How to mix burnt umber with the 3 primaries and black/white
To get a kind of burnt umber, I think (I can't test it now) you need around 3 parts black, 3 parts red, 1 part blue and 1 part yellow.

Let's assume the first picture I found googling for fur is the kind of reference you have:



For painting fur, and especially to get a warm glow in your painting, I suggest painting a layer of red (with yellow mixed in as desired). Let it dry. This will be your base layer.
Mix some black and red (or use the 'burnt umber' recipe, as the cooler colour will contrast nicely with the warmer one, emphasizing it), and dilute it with water, as you want the next layer to be quite transparent (at first, anyway). Apply this mix to the cured base layer, using a brush to create the texture you want. You can change the proportions of the solution to get darker and lighter areas.  

Alternatively, you can apply a much darker, semi-transparent layer on top of the warm red base layer, giving you the dark area as seen in the image to work up from. Using a small brush, you can paint in the strands to get a more detailed picture, mixing it with white to get the sheen, which is quite important for the realistic appearance of any kind of hair.