That's sort of what I'm doing, building my Square Foot boxes on top of 2 laysers of thick cardboard right on top of unused (but mowed) pasture. Eventually that cardboard will disintegrate, but hpefully that grass and any weeds will be thoroughly dead by then. I tried to kill it by putting a black tarp (dark brown actuallly) over it and leaving it for a month. Ha! It just became a mini green house. The grass grew just fine underneath, it just was a paler grean. Since we were in a drought at that time, the grass seemed to like having the black tarp over it -- held in the moisture!
I gathered up the tarp and moved the garden to a more convenient location (nearer the gate -- I'm not totally stupid) and tried the cardboard/weed barrier method. Since there is no planting mix in the boxes yet, I may add newspaper at the bottom too. In any case, the grass in my previous location bounced back and, within a week, looked like nothing had been there at all!
I even have cardboard under the "plastic mulch" that will be the aisles. I do not want these beds to be contaminated by grass clippings when I mow, as much as I would like to have grass aisles. That would not be possible for me to maintain.
Just went out and “dug the potatoes”....took less than five minutes, just pulled back the straw and put them in a basket....Nice part about it there were no damaged ones from a spade or shovel. Will use the same technique next year .....only larger. Start saving those papers/cardboard now!