Video: America has a long and storied history with regulating alcohol, which has left them with a high drinking age and a very strange requirement that all alcohol be slightly radioactive. As Tom Scott explains, the requirement stems from the fact that all alcohol sold for human consumption has to be made from plant matter -- no booze made from oil allowed. To enforce the law, the Alcohol and Tobacco Tax and Trade Bureau (TTB) use radiocarbon dating -- recently-deceased plant matter will have a much higher level of carbon-14 than the distilled product of millenia-old oil.
So technically, the rumour is correct: all alcohol has to be slightly radioactive in order to be legal in the US. But it's not going to kill you any time soon, and in case you're worried about drinking shots while on your backpacking trip across the States, just remember that your body also contains radioactive carbon-14.