Been there, done that!
* * W A R N I N G! * * The biggest concern is the label side because some of the coatings that were used will be removed if done improperly. This means not only the silkscreened label but also the aluminium refective layer too. Once the reflective layer is gone, the cd becomes a glass coaster.
I did my cds (~ 300 about 10 years ago) and ended up losing three cds. The trick is to warm up the ring with a hair dryer on the lowest setting before picking at it with your fingernail. As it starts to lift off, continue to apply heat and monitor removal progress to make sure it will come off. I did get all of mine off. A royal pain to do because it's a slow & easy effort.
Yes, it will leave a residue and goo gone or lighter fluid will take it off. However, some of the silkscreening ink may come off too.
If I were you, I would burn copies of all of them before removing rings. Burning was not available to me when I did my removal project and it took me years to repurchase the ruined cds because they were OOP cds.