Radioactivity of materials
No. Radioactivity means that the nucleus of a particular atom is unstable, and the nucleus will decay to a more stable state either by decomposing or emitting energy (as a photon). This instability will not make other nuclei unstable unless the decay of one of the radioactive atoms causes a fission process in one of the atoms in the desk. This is a secondary effect though and surely will not happen once the material is removed.

Another way of thinking about this is that there are materials in which people house radioactive substances. These materials do not become radioactive or else there would be no point in having them in the first place. Not just any material will work though because radioactive emission products might be transparent to many materials or are energetic enough to melt/decompose other materials. So, I would bet the desk would get destroyed, but from normal decomposition not from decay.

Finally, the reason that it seems like things become radioactive once there has been radioactive material nearby is because we usually think about the products of nuclear fission. These products will have quite long half-lives and thus the material will stay around for a long time decaying.