I only tape my Nesties down if I'm wanting to particularly cut out a certain part of my cardstock - like if I've done a Distress background and I like the way one area came out or if I'm working with a pattern that needs to be dead square. Or if I'm cutting a window in a card front.
I use blue painters tape. You can run a small piece right across the die on one or both sides to hold it in place. It comes right off the die cut piece at the other end and I have never had any gunk build up on the die. I usually take the bits of tape and stick them to either side of the top of my Big Shot to reuse.