Should I bind the edge of an unbound remnant rug? If so, how do I do so?
You absolutely should (must) bind the edges. 

Many consumer rugs are cut to length from a roll stock and bound for sale. Binding the edge is how it transforms from a textile on a roll into a rug.

You could do it yourself if you had the requisite equipment and sewing skills but I can't imagine it would be tremendously expensive to have done. 

You can find a video of a product that can be used to bind your own edges, as well as basic instructions on how to use it right here.

A summary of the video is below:


  Tools you'll need
  
  
  Hot glue gun
  Scissors
  Straight edge
  
  
  Procedure
  
  Start by trimming the rug to provide a clean smooth edge.  You can do this using a straight edge and a pair of scissors.
  
  Next you'll want to trim the binding, so you start with a clean straight end.
  
  Pull back the backing paper, and apply the binding starting at the middle of one edge.
  
  When you get to the corner, use the edge of the rug as a guide to make a cut through the flat part of the binding. Then bend the binding around the corner, and continue around the rug.
  
  Once you've reached the place you started, use the beginning end as a guide to cut the binding to the proper length. Trim any excess strings, then attach the ends using hot glue.
  
  Run a 1/8" bead of hot glue, between the edge of the rug and the piping. Do this in small sections, so you can hold the two together until the glue sets. Continue all the way around the rug. 


Alternatively, any rug shop or commercial carpet establishment should be able to do it for you...