No, radioactive dating does not damage a fossil. This is because it only takes a small proportion of the bone and determines the isotope at a molecular level. This is why it is much more convenient and easy to use radioactive dating.
Not only does it preserve the fossil but it also allows the efficient dating of the sample with accurate estimation of the age of the fossil. This has made radioactive dating a milestone for geologists enabling them to date rock samples as well as fossils and other remains. The method was developed in 1990 and since then has enabled geologists to efficiently calculate the age of the specimen.