Different colors of painters tape have different levels of stickiness. White is the most sticky, the blue is less so. The green tape used on these cards is far less sticky, and would pull off paper easily without damaging it.

What a very cool card!  Can't wait to make one  (with everyone's altered suggestions of course)

Hey, maybe that's how you could do it....stick the tape on some fuzzy fabric first to make it less sticky. 

Tony--You'd think so, right?? I've had some serious problems with it, however. I painted the interior trim on a rehab we did a few years ago. I taped off the walls before I did it, of course, and then when I removed the tape, half the dryway came with it. (Hyperbole.) It was the good 3M tape, so I thought I was safe. For the rest of the job, my solution was to stick the tape on the floor, carpet, my jeans, etc. before applying it to the wall to make it less sticky. What a pain!

CraftyKat's Post-it idea seems right. They do make repositionable adhesive spray. Maybe you could spray that on strips to use as fake tape, apply it, then paint over the whole thing. When it's pulled back, all is revealed. 

It could be Post-it TM tape or a type like that.  I've used it to tape off the edges of wallpaper border so I could repaint part of a damaged wall.  It's just like a Post-it TM note, but on heavier white paper.  I've never seen it in green, though.

Well, it's painter's tape, so it's not all that sticky right?