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Look close you will see common spearmint growing thought the burlap that I mulched a blue berry plant with. The burlap is doubled because it is a sack & I left it & just layed it in the bed.The burlap is 36 months old, It will work on most, if not all annuals, but not on all perennials.I get the bags free from a coffee plant, so I use them a lot. The bags can be stacked 3 high if you over lap them by 3 inches & cover them with straw or leaves, to filter the sun light.

Both burlap and the coconut fiber are too loosely woven to be effective "weed" barriers. They will allow enough light to reach any seeds that germinate under them so those will grow. That material is used on newly seeded turf areas to help hold the seed in place and the turf grass grows right through it.
To effectively kill the grass growing there you need a barrier that will exclude all light, something like newspaper or cardboard, which then would need something to hold it in place.

A layer of 3 bags will stop all annuals(seeds & roots).
The bags in the pic are old & rotten, so the mint had no problem pushing though. I can stop perennial grasses for a year, some will die, others will go though or around.
Some can live under 1 inch of cardboard after Everything is dead around them.
But the company I work at sprays round up on these grasses & they die back, then come back from the root. So wood,cardboard,burlap & 6 inches of compost & mulch do not stand a chance.
But the burlap is free & easy to put down will stop 80% by it's self & as I stated above with mulch you will be happy with it.