My sweetie has used a product that is a bit like glue to prevent fabric from Fraying.
she has used it on fine delicate fabrics...not sure if it is robust enough for burlap. You could always wrap the edges in a ribbon like material and stitch them.
Kevin is correct in what the Aleene's product will do, but it will show a little. If you don't want that your best bet is to pull a thread down the center of the fabric lengthwise where you want to cut. Once that is accomplished run a long zig zag stitch in a color of thread that closely matches the color of burlap down each side of the the missing thread. Once that is finished you can cut along the missing thread.
Though pulling the thread is not absolutely necessary, it will help you greatly in seeing where to sew. Beware that tiny fibers will fly so if you have allergies this could affect them. Wear a mask and protective glasses if that's the case.