If a sample to be carbon-dated is wrapped in a piece of paper, will you still be able to effectively measure the radioactivity of the sample?
Hardly, with or without paper wrap.

Most of the radiation will be absorbed by the sample itself, you'll have a hard time modelling the absorption to derive the actual remaining activity of the carbon. Artifacts are usually not a piece of completely homogeneous material. Also you'll have a hard time finding anyone with a suitable Geiger counter, if you want to do more than guesswork.)

Modern radiocarbon methods are destructive, you take a number of tiny (&lt;1 mg), representative samples out of your artifact, and directly do mass spectrometry on them, without waiting for radioactive decay.