How to remove airlines/security tag stickers from suitcases?

I used White Spirit on my Polycarbonate luggage and it worked very well. No damage to luggage.

Just removed plastic document sleeves with sticky back plastic which were stuck on expensive suitcases.
Used a hair dryer to heat up the plastic then slowly and carefully peeled, removing the lot in one go by teasing the plastic away from the case (this took about 10 mins – A4 size sticker.
I then used regular olive oil to remove the sticky residue left behind, let it soak in for a few seconds before rubbing off. The suitcases are back to normal with a new shine finish!

Food Safe Mineral Oil to protect kitchen woodenware works great and doesn’t remove any coating…I just tried it on my new Samsonite luggage

I have successfully removed labels using water and a cloth to remove the paper exposing the adhesive than using white spirit on a paper towel to remove the adhesive.

Converted from a comment to this answer. Since we are all sharing Grandma’s secret stain remover tricks: in my experience, gasoline (AE, petrol in British English) works much better than alcohol to remove adhesive residuals.

Heat the areas to a point where the glue melts but the plastic doesn’t, then slide the sticker off. Best things to do this with are a blowtorch, some hot coals, an open fire or a really good heater.

Alternatively bathe your suitcase for a few hours (perhaps even overnight) and then put it somewhere nice and warm to dry off.

Remove them as soon as possible, they become harder to remove over time.

Use a hair dryer to warm up the stickers, this “softens” the glue (at least for some types).

Pull very gently so as not to tear the stickers apart, and give the glue time to separate.

As a last resort, use alcohol, acetone or another organic solvent to dissolve or weaken the glue.

I’ve found that using medical adhesive remover (like this one) works to remove stickers.

I happen to have a bunch lying around that I’ve never used, but you can order them from Amazon (they’re not expensive; if you look, you’ll probably find something cheaper than I did).

Nail polish remover, of which the principal ingredient is Acetone, is a general purpose solvent, and works well on many organics such as glues.
If you have diabetes, especially type 1, your urine might work too (due to Acetone).
You might want to check that it won’t dissolve your suitcase too.

I think labels vary in their adhesive systems employed, so there is no one ‘right’ answer. I can’t speak for the proprietary products mentioned in other answers, as I tend to avoid spending money on these when a common chemical will do the trick, but trying the following should remove most labels:

1. Peeling carefully and slowly – works sometimes with adhesives that don’t bind permanently to plastics. The sooner after label application the better, before the adhesive ‘closes’
2. Moisten with water, hotter the better.
3. Moisten with alcohol
4. Moisten with white spirit (or if desperate, gasoline)
5. It is unlikely that any of the above will damage what’s below the label – acetone sounds a bit drastic.
6. Abrasion with rubber or fingers.

Olive oil worked well for me. Just let the sticky residue soak in it for a bit before rubbing it with a cloth.

Another product I have tried is a sort of a cleaning eraser. It works well, but it also requires quite a bit of rubbing

(source)

I have yet to meet a label that didn’t yield easily to Goo Gone. Beware that I’ve found some printing on packaging that also wipes off with it.

I find this stuff works extremely well on any paper labels (not so well on plastic labels) …

… available from many places (e.g. Maplins in the UK and from any Servisol stockist)

You can use very hot water, the coarser (dark green) side of a dishwashing sponge, and quite a bit of elbow grease to scrub the sticky part of the tape/stickers away. This might be a very time consuming operation depending on how many such stickers you have to clean off.

If the suitcase is tough, and you are not afraid to scratch it, you can increase your effectiveness by using either the back of a box cutter blade (left) or what is called a Venetian plaster spatula (right) to scrape away the stickers (images from Amazon and Wikipedia):

One thing I use and are available easily is, ahem, Alcohol. A few drops and rubbing tight can clean the stickers very easily

There are many alternatives too. WD-40, anything with acetone, Goo Gone, etc.

As Dan Neely points out, be cautious when using acetone because it can melt some plastics.