How can I stop dye from rubbing off burlap?
It is hard to guess what went wrong in the dying process, but I am assuming you have a natural jute fiber burlap. You will probably have to dye the fabric again. What usually works is Rit - which you can get at pretty much a y fabric store. Be careful, it is a little caustic and will stain your skin.

If you want to try to fix the dye, don’t use salt or vinegar - caustic soda might work, but you probably want to get retayne as it is described here. Apparently also available at quilting and fabric stores, I must say that I do not know the product but the science seems sound.