How do I lift a fitted carpet without damaging it?
You can pull carpet up off the tacking strip to do some work. It depends on how much of the carpet you're going to pull up. Carpet is stretched over tack strips using a kick stretcher or a power stretcher so if you pull the WHOLE carpet it you risk losing that stretch. I did some work under our carpet and wound up pulling up too much and had to hire a company to come out and re-stretch and tack it back down but if you're just pulling up a small corner you should be ok to just push it back down.

(Word of warning: when pushing the carpet back down on the tack strip, use a board to push it down, the tacks on the strip routinely go through the carpet and it hurts like all get out if it suddenly jams into your fingertip)

If the work is in the middle of the floor I've used a utility knife and made a lengthy slit in the carpet to do what I needed to and then called out a carpet install company to re-seal the carpet with a glue strip. After he re-glued the carpet and ran the seam blending tool over it it looked perfect. I would highly advise against nailing through the carpet, it creates an indent and doesn't look very good.