I use lighter fluid on a paper towel to wipe the goop off, then wipe it off well with a dry one and you'll have no damage at all.  There are a couple of thrift stores here that attach price stickers to their books and they don't peel well without leaving some residue----at least that's often the case.  The lighter fluid solvent works well for this.

If you don't have lighter fluid, hairspray sprayed on a paper towel is good too.  Also, alcohol, which is what hairspray has in it.

I like to use hand sanitizer. It's got enough alcohol to make it dry before it wets the book cover, but not as flamable as lighter fluid, and hair spray can sometimes get sticky. I picked up a cheap small bottle from the samples aisle at Walmart, and saw them in the drugstores too.

There are products made for this.  One is called "goo gone" the other is "undo" or "undu."  Both are yellow in color just so you know what you are looking for.  You should be able to find either at Target, Wal Mart, etc.

I love a product called "De-Solv-it " made by OrangeSol--For sticky adhesive residue,  I use a teensy, tiny drop of De-Solv-it and rub with a cotton t-shirt or white terry cloth- I don't like paper towel and tissue because it crumbles and leaves little pieces of paper - I can purchase De-Solv-it at Smart & Final, Storables, or Ace Hardware- I'm sure it is sold at more places- 

Also--For smudges and dirty spots , I clean up books with an  art gum eraser or a white plastic eraser-

I use an alcohol wipe to gently wipe the sticky and then I dry the cover good with a Kleenex.

I use alcohol wipes.  They seem to work the best for me.....Also, goo-gone is some good stuff....

I've had success with Goo Gone.

Lighter fluid here.   Never had it fail, never had a book damaged.

I did it once with rubbing alcohol on a paper towel:  Peel off as much of the sticker as you can, then rub on the alcohol, then rub hard with a dry paper towel before the alcohol evaporates.  If that doesn't get it all, repeat.  I think I had to do it 2-3 times.  Cover looks as good as new and has no signs of sticker gum, nor signs of the liquid alcohol soaking in.