You'd need a way to keep the straw completely dry. Also a way to keep the chickens from bumping into and pecking at the burlap thus releasing huge clouds of dust and whatever mold grew from incomplete dryness. Also also, a way to be really darn sure that mice can't get into the straw where they will otherwise make a merry mess and turn into ten thousand mice. I'm not sure these things are going to be real do-able.
If you are really set on using straw and no plastics, what about stacking a bunch of straw bales all around the OUTSIDE of your coop, up to the eaves almost. Mice will still move into the straw and things will still get moldy and dusty, but it'll basically be on the OUTSIDE of the coop, where it won't bother the chickens so much.
Good luck,
Pat