How to get sticky residue off plastic-covered furniture?
Generally one experiments with various cleaning products and escalates when the initial choices fail to work. i.e. neither you nor we know exactly what the residue 15 years of plastic wrapping has left on your chair's finish is, so we start with things that are normal for cleaning and unlikely to damage the finish, then move on to products that are generally "more aggressive" but which also require testing to be sure they don't damage the finish.


Furniture polish
Spray cleaners
Cleaning wipes


Solvents in no particular order (test on a part of the chair that won't show to be sure they don't damage the finish - for instance, alcohol will dissolve shellac, but shellac is rather unlikely unless you have seriously antique chairs.)


Denatured alcohol 
mineral spirits/paint thinner
naptha
lacquer thinner