What chemical is responsible for the distinictive odor of Scotch tape?
This is a long answer, but according to this:  


  While some pressure sensitive tapes are still prepared with natural
  rubber, the majority are now made using mostly synthetic materials.
  The backing for cellophane tape usually consists of cellulose acetate,
  a synthetic derivative of cellulose, which comes from wood pulp or
  cotton seeds. The cellulose is chemically treated with acetic acid and
  anhydride, and the side that won't receive an adhesive coating is
  treated with a release agent that enables the tape to be wound and
  unwound without sticking together. Although this compound varies among
  manufacturers, some commonly used substances include stearato chromic
  chloride and polystearic carbonate. Prior to the application of the
  adhesive, the adherend side of the backing may be primed with a
  solvent or aqueous dispersions such as nitrile rubber or chlorinated
  rubber. To produce the final adhesive substance, some manufacturers
  use as many as 29 raw materials that go through various stages of
  production. However, the generic adherend is made up of acrylic
  resins, petroleum byproducts that are broken down into alcohols and
  acids before being fused into a polymer compound. This compound is
  then mixed with mineral spirits or a hydrocarbon solvent, creating an
  aqueous emulsion (a solution in which the microscopic resin particles
  are held suspended) that is applied to the backing.


In other word, there are many candidates.  I consider the top possibilities of those listed above to be: compounds composing the cellulose, the nitrile-based rubber compounds, and the semi-volatile organics composing both the adhereant and the base material.  With so many compounds going into a single product, it is likely the mixture that gives the distinctive smell rather than any individual compound.