Parenthackers who sew are probably aware of this tip, but it might be useful for everyone else.
Seam rippers allow you to quickly remove itchy tags (or appliques) from clothes without leaving pointy edges or little holes behind. Basic rippers are often available in grocery stores (next to the button repair kit) for about $2, or you can get fancier versions at craft- or sewing supply stores.
(Brings back memories of playing with the notions in my mom's sewing box.)
Another Parenthacker deals with annoying shirt labels by tacking them down with iron-on hemming tape.
How do you deal with scratchy clothing tags?