How to treat the edge of OSB in a piece of furniture
I'd tend to fall in the "OSB furniture sounds like a terrible idea" camp, but anyway...

Two steps come to mind - one is to seal the cut edges thoroughly using either a thin epoxy or a varnish/polyurethane type of product, to both bind small slivers and prevent or at least slow moisture intake and resultant edge swelling. The other is to sand the edges (you want to seal first, then sand so that sanding doesn't turn into peeling off slivers) - and you may want to seal again after sanding.