It DOES result in radioactivity: plain old "gamma radiation" regardless what you use as fusion fuel.
A few days old, but I felt this needed to be addressed. Nerding ahead, steer clear if you want to avoid it...
Gamma radiation is not radioactivity. It is basically just ionizing radiation that consists of high-energy photons that can be borne from pretty much anything that has charged particles and is energetic enough.
True, radioactivity is one of the most common phenomena that produces ionizing radiation - which counts not only gamma, but also alpha- and beta+/- -radiation types, which unlike gamma are actually charged particles shooting from the nuclear reactions. Alpha particles are Helium 4 nuclei; beta- particles are electrons and beta+ particles are positrons, or antielectrons if you prefer that.
Radioactivity is basically an archaic and rather unsuitable term from Curies' days of research, describing that something's going on in the subject matter and probably they noticed it from the accelerating charged particles generating noise heard by radio or something along those lines, I don't remember the actual history of the discovery of radioactivity. I have no idea why the term radioactive has remained popular, when simple "active" is actually better describing term.
Basically, radioactive matter is stuff that has active isotopes in it - atoms that are have unstable nuclei and might or might not disintegrate any given moment. The probability of an individual disintegration stems such statistical factors like half-life and the activity of the matter, but that's not really important.
Now, there are ways to make stable matter active, by unstabilizing part of the nuclei in it. By far the easiest way to do it is to bombard the matter at hand (figuratively, hopefully not literally at hand) with neutron flux. When the neutrons hit the nuclei, they kinda assimilate themselves as parts of the nucleus they have hit, which increases the atomic mass of that particle nucleus and may or may not make the nucleus active, that is "prone to disintegrate".
When you bombard any material with neutrons long enough - say, a couple dozen years next to a blazing nuclear reaction that releases a lot of neutrons - it will slowly turn into variably active nuclear waste, regardless of what kind of radiation is released in the primary reaction used in the power plant.
However, if you manage to use a reaction that releases only little or no free neutrons, the parts of the reactor do not become active. Basically, the less neutrons released, the better for everyone - then the only ionizing radiation from the reaction would be the primary radiation, which in fusion's case is indeed gamma radiation. However, shielding the surroundings from primary radiation is a mundane task and just requires a lot of mass around the reactor. The other stuff borne in the reactions - like the resulting nuclei, which are by nature charged particles - can be used as source of thermal energy to run turbines (ye olde way) or they can be converted into electrical current via magnetic fields and induction (which would be the better and more efficient way to produce electricity from the reaction).
