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.