While it is true that some fossils can be radioactive, not all of them are. According to the ressources I have found, it depends on the place where these minerals are found and also the other minerals present at this place.
When there are uranium and or potassium rich ores available, the fossils seem to enrich these radioactive isotopes in the calcium matrix of the bone during fossilation. See the reference for some more details on this.
The isotopes which are occurring there are Uranium (U235 and U238), Potassium (K40)) and Rubidium (Rb87), all of them with half lives in the billion years. Carbon does not play a role in these very old fossils (millions of years old), as the half life of the relevant carbon isotope is to short. These elements are also used for radiodating these fossils, see the table below (from here):
Regarding the radioactive safety I think there are some fossils where you should be cautious and carry them around all day long in your pocket. Most of these isotopes release $\alpha$ and $\beta$ radiation, some $\gamma$. While the $\alpha$ radiation is relatively easy to shield, lead can have detrimental effects due to the occurance of retardation radiation of the slowed down particles. In the lab we use thick acrylic glass shields to protect us from $\beta$ radiation.
Additionally I have found some secondary sources for painting the dinosaur bones with lead paint, but no real first hand source from a museum or so. Additionally the rules for radiation safety are pretty strict. If the bones would be so radioactive that a short visit would endanger you (how long are you looking at such pieces, a few minutes at maximum I would say) than they couldn't be shown anyway. Let alone the dangers for the scientists which analyze them. So I would think this is an urban legend.