Usually it's neutron radiation that makes things radioactive[1]. Really energetic particles can do the same[2]. But the positron emission from Flourine-18 I don't think has enough energy to do that.
[1] Being neutral particles they can easily be absorbed by the nucleus of whatever. Adding an extra neutron may result in an unstable nucleolus.
[2] Friend that worked at SLAC (big linear accelerator) gave us a tour and mentioned the block of aluminum they use to backstop the beam. He said it gets slightly radioactive.
It's sort of like the idea that a microwave can be dangerous running exposed, but it's fine the moment the magnetron stops.
If you have a rapidly decaying isotope, it will produce lots of radiation. However, most of that radiation will be harmless.
Some ionizing radiation will be able to change the atoms that make up surrounding materials, but most of that will remain harmless, and very little will become radioactive.
Wouldn't it radiate into the room and make the room... bad? (Sorry, I'm clueless about this subject)