Hang posters with straight pins, with sticky-side-out spirals of masking tape or with adhesive dough. For framed pictures, use the smallest nail you can get by with and then buy spackling compound to carefully fill the holes before you move.
Cover furnished sofas or upholstered chairs with a washable throw, quilt or washable slipcovers if you eat in the living area.
Put an inexpensive throw rug - preferably washable - or a disposable carpet remnant under the table if the dining area is carpeted. Use a special pad to keep the top rug flat.
Purchase washable, tie-on chair cushions if the chair seats in a furnished apartment are covered with fabric.
Ask permission before you paint. Chances are, the landlord's answer will be no or will be that you may only paint the walls off-white; save enough extra paint for wall touch-ups.
Hide bad walls (even gouges in the drywall) with posters if you haven't been given permission to paint and the landlord won't paint or make cosmetic repairs.
Hang curtains with spring-tension rods that mount inside the window. You won't have to fill and touch up nail or screw holes from curtain-rod hardware.
Put place mats or other protective gear under tabletop furnishings such as lamps and electronics gear to prevent scratches.