Mr. t. I've pulled down headliner BUT it was fairly new and not degraded or damaged like the small pieces underneath which I suspect were original. IF you're very careful, it can readily be done. Source a thin tip, wider blade screwdriver for prying out the old staples AND a good set of needle nose pliers for those half-out stubborn ones. Try not to stretch the existing liner if you're only removing part of it. Tape it up out of place if necessary. Of course, SS staples to re-fasten.
On second thought, I was dealing with vinyl headliner and our former MT had solid panels with vinyl glued on it so the technique will be different. IF removing a panel, support the center of it so it doesn't sag and put undo pressure on those last screws/nails you will be removing.