There are some isotopes which decay naturally by emitting neutrons, but this is very rare so it is usually ignored in any discussion I give about radiation. However, in a nuclear reactor, there are many neutrons. While these neutrons are what keep the fission chain reaction going, they are also what make the reactor vessel radioactive. In the case of food irradiation, neutrons are not being used. Alpha radiation does not cause this sort of behavior because the alpha particles are so massive and highly charged that they slow down by ionizing the atoms around them, not by interacting with nuclei.
Beta particles are similar since they also have charge.
They tend to ionized atoms and interact with electrons. Finally, gamma radiation has three main types of interactions, all involving interactions with atomic or ionized electrons. Thus, without having interactions with nuclei which change their composition, this transmutation will not be possible. Since the food is not radioactive originally, it will not be radioactive after irradiation.