A lot of people, (including me with my burlap table runner), have problems with the burlap being smelly or wrinkly. So I will tell you what I did to take care of the smell based on tips I have used from the internet. These may not work for you, so don't take my words as if I'm an expert on the issue.
What you can do, to make sure you don't ruin ALL of the fabric, is to take a little snippet and test out your cleaning methods on that. This is a tip I have seen come highly recommended around the web - and it makes a lot of sense.
Now, onto the recommendations:
The place I got the burlap from recommended to me that I gently sprinkle baking soda over it to remove the smell.
Other simple tricks and tips are to shake it out, smooth it out and then let it sit outside in the sun. Heat works really well in that it helps to remove the smell somewhat.
Ironing it - similar to the trick of putting it in the sun - you will smooth out the burlap and alleviate the smell a little bit at the same time.
Washing it by hand - this won't work so well if you have a lot of burlap, but it is better than machine washing it which may cause the burlap to become ruined.